---
# System prepended metadata

title: 機率與統計公式

---

Probability & Statistics for Engineers & Scientists
===

# 積分公式

## 分部積分
$$
∫udv=uv−∫vdu
$$

# Chapter 1

## Definition 1.1
>平均

Suppose that the observations in a sample are $x_1, x_2, . . . , x_n$. The **sample mean**, denoted by $\overline x$, is
$$
\overline x =\sum\limits_{i=1}^n\frac{x_i}{n}= \frac{x_1+ x_2+ . . . + x_n}{n}.
$$

## Definition 1.2
> 中位數

Given that the observations in a sample are $x_1, x_2, . . . , x_n$, arranged in **increasing order** of magnitude, the sample median is
$$
\tilde{x}=
\left\{ 
  \begin{array}{l}
    x_{(n+1)/2}, & \text{if n is odd}\\ 
    \frac{1}{2}(x_{n/2}+x_{n/2+1}), & \text{if n is even}
  \end{array}
\right.
$$

## Definition 1.3
> 變異數

The **sample variance**, denoted by $s^2$, is given by
$$
s^2=\sum\limits_{i=1}^n\frac{(x_i-\overline{x})^2}{n-1}.
$$

> 標準差

The **sample standard deviation**, denoted by $s$, is the positive square root of $s^2$, that is,
$$
s=\sqrt{s^2}.
$$

# Chapter 2

## Definition 2.1
The set of all possible outcomes of a statistical experiment is called the **sample space** and is represented by the symbol $S$.

## Definition 2.2
An **event** is a subset of a sample space.

## Definition 2.3
The **complement** of an event $A$ with respect to $S$ is the subset of all elements of $S$ that are not in $A$. We denote the complement of $A$ by the symbol $A'$.

## Definition 2.4
The **intersection** of two events $A$ and $B$, denoted by the symbol $A \cap B$, is the
event containing all elements that are common to $A$ and $B$.

## Definition 2.5
Two events $A$ and $B$ are **mutually exclusive**, or **disjoint**, if $A \cap B = \phi$, that
is, if $A$ and $B$ have no elements in common.


## Definition 2.6
The **union** of the two events $A$ and $B$, denoted by the symbol $A \cup B$, is the event
containing all the elements that belong to $A$ or $B$ or both.

## Definition 2.7
A **permutation** is an arrangement of all or part of a set of objects.

## Definition 2.8
For any non-negative integer $n$, $n!$, called "$n$ factorial," is defined as

$$
n!=n(n-1)...(2)(1),
$$

with special case $0! = 1$.

## Theorem 2.1
The number of permutations of $n$ objects is $n!$.

## Theorem 2.2
The number of permutations of $n$ distinct objects taken $r$ at a time is

$$
C_r^n=\frac{n!}{(n-r)!}
$$

## Theorem 2.3
The number of permutations of n objects arranged in a circle is $(n-1)!$.

## Theorem 2.4
The number of distinct permutations of n things of which $n_1$ are of one kind, $n_2$ of a second kind, . . . , $n_k$ of a $k$th kind is

$$
\frac{n!}{n_1!n_2!...n_k!}
$$

## Theorem 2.5
The number of ways of partitioning a set of $n$ objects into $r$ cells with $n_1$ elements in the first cell, $n_2$ elements in the second, and so forth, is
$$
\begin{pmatrix}
  n\\
  n_1, n_2, ...n_r                
\end{pmatrix}
=\frac{n!}{n_1!n_2!...n_r!}
$$
where $n_1 + n_2 + · · · + n_r = n$.

## Theorem 2.6
The number of combinations of $n$ distinct objects taken $r$ at a time is
$$
\begin{pmatrix}
  n\\
  r                
\end{pmatrix}
=\frac{n!}{r!(n-r)!}
$$

## Definition 2.9
The **probability** of an event $A$ is the sum of the weights of all sample points in $A$. Therefore,
$$
0 \le P(A) \le 1, ~~~P(\phi) = 0,~~~ \text {and}~~~ P(S) = 1.
$$
Furthermore, if $A_1 , A_2 , A_3 , . . .$ is a sequence of mutually exclusive events, then
$$
P(A_1 \cup A_2 \cup A_3 \cup· · ·) = P(A_1) + P(A_2) + P(A_3) + · · · .
$$

## Theorem 2.7
If $A$ and $B$ are two events, then
$$
P(A \cup B) = P(A) + P(B) − P(A \cap B).
$$

## Theorem 2.8
For three events A, B, and C,
$$
P(A \cup  B \cup  C) = P(A) + P(B) + P(C) − P(A \cap  B) − P(A \cap C) − P(B \cap C) + P(A \cap B \cap C).
$$

## Theorem 2.9
If $A$ and $A'$ are complementary events, then
$$
P(A) + P(A') = 1.
$$

## Definition 2.10
The conditional probability of $B$, given $A$, denoted by $P(B|A)$, is defined by
$$
P(B|A) = \frac{P(A \cap B)}{P(A)} ,~~~ \text {provided}~~~ P(A) > 0.
$$

## Definition 2.11
Two events $A$ and $B$ are independent if and only if
$$
P(B|A) = P(B) ~~~\text {or}~~~ P(A|B) = P(A),
$$
assuming the existences of the conditional probabilities. Otherwise, $A$ and $B$ are **dependent**.

## Theorem 2.10
If in an experiment the events $A$ and $B$ can both occur, then
$$
P(A \cap B) = P(A)P(B|A),~~~ \text{provided}~~~ P(A) > 0.
$$

## Theorem 2.11
Two events $A$ and $B$ are independent if and only if
$$
P(A \cap B) = P(A)P(B).
$$
Therefore, to obtain the probability that two independent events will both occur, we simply find the product of their individual probabilities.

## Theorem 2.12
If, in an experiment, the events $A_1,A_2, . . . , A_k$ can occur, then
$$
P(A_1 \cap A_2 \cap· · · \cap A_k )
= P(A_1)P(A_2|A_1)P(A_3|A_1 \cap A_2 ) · · · P(A_k |A_1 \cap A_2 \cap· · · \cap A_{k−1} ).
$$
If the events $A_1,A_2, . . . , A_k$ are independent, then
$$
P(A_1 \cap A_2 \cap· · · \cap A_k )= P(A_1)P(A_2). . .P(A_k)
$$

## Definition 2.12
A collection of events ${\oldstyle{A}} = \{A_1, . . . , A_n \}$ are mutually independent if for any subset of ${\oldstyle{A}}$, $A_{i_1} , . . . , A_{i_k}$ , for $k \le n$, we have
$$
P(A_{i_1} \cap · · · \cap A_{i_k} )= P(A_{i_1})· · ·P(A_{i_k}).
$$

## Theorem 2.13
If the events $B_1,B_2, . . . , B_k$ constitute a partition of the sample space $S$ such that $P(B_i) \ne 0$ for $i = 1, 2, . . . , k$, then for any event $A$ of $S$,
$$
P(A) = \sum_{i=1}^kP(B_i \cap A) =\sum_{i=1}^kP(B_i)P(A|B_i).
$$



## Theorem 2.14
**(Bayes’ Rule)** If the events $B_1,B_2, . . . , B_k$ constitute a partition of the sample space $S$ such that $P(B_i) \ne 0$ for $i = 1, 2, . . . , k$, then for any event $A$ in $S$ such that $P(A) \ne 0$,
$$
P(B_r|A) = \frac{P(B_r \cap A)}{\sum_{i=1}^kP(B_i \cap A)}= \frac{P(B_r)P(A|B_r)}{\sum_{i=1}^kP(B_i)P(A|B_i)}~~~
\text{for}~~~ r = 1, 2, . . . , k.
$$

# Chapter 3

## Definition 3.1
A **random variable** is a function that associates a real number with each element in the sample space.

## Definition 3.2
If a sample space contains a finite number of possibilities or an unending sequence with as many elements as there are whole numbers, it is called a **discrete sample space**.

## Definition 3.3
If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a **continuous sample space**.

## Definition 3.4
The set of ordered pairs $(x, f(x))$ is a **probability function**, **probability mass function**, or **probability distribution** of the discrete random variable $X$ if, for each possible outcome $x$,

1. $f(x) \ge 0,$
2. ${\sum\limits_{x}} f(x) = 1,$
3. $P(X = x) = f(x).$

## Definition 3.5
The **cumulative distribution function** $F(x)$ of a discrete random variable $X$ with probability distribution $f(x)$ is
$$
F(x) = P(X \le x) = {\sum\limits_{t\le x}} f(t), \text{ for $−∞<x<∞$}.
$$

## Definition 3.6
>機率密度函數

The function $f(x)$ is a **probability density function** (pdf) for the continuous random variable $X$, defined over the set of real numbers, if
1. $f(x) \ge 0$, for all $x \in R.$
2. $\int_{-∞}^∞f(x) dx = 1.$
3. $P(a < X < b) =\int_{a}^bf(x) dx.$

## Definition 3.7
>累積分佈函數

The **cumulative distribution function** $F(x)$ of a continuous random variable $X$ with density function $f(x)$ is
$$
F(x)=P(X \le x) =\int_{-∞}^xf(t) dt, \text{ for $−∞<x<∞$}.
$$


## Definition 3.8
>聯合機率分佈、機率質量函數

The function $f(x, y)$ is a **joint probability distribution** or **probability mass function** of the discrete random variables $X$ and $Y$ if
1. $f(x, y) \ge 0$ for all $(x, y),$
1. ${\sum\limits_{x}}{\sum\limits_{y}}f(x,y)=1,$
1. $P(X = x, Y = y) = f(x, y).$

For any region $A$ in the $xy$ plane, $P[(X, Y ) \in A] ={\sum}{\sum\limits_{A}}f(x, y)$.


## Definition 3.9
>聯合密度函數

The function $f(x, y)$ is a **joint density function** of the continuous random variables $X$ and $Y$ if
1. $f(x, y) \ge 0$ for all $(x, y),$
1. $\int_{-∞}^∞\int_{-∞}^∞f(x,y)dxdy=1,$
1. $P[(X, Y ) \in A] =\int\int_{A}f(x, y)dxdy$, for any region $A$ in the $xy$ plane.

## Definition 3.10
>邊際分佈

The **marginal distributions** of X alone and of Y alone are

$g(x)={\sum\limits_{y}}f(x,y)$ and $h(y)={\sum\limits_{x}}f(x,y)$, for the discrete case

$g(x)=\int_{-∞}^∞f(x,y)dy$ and $h(y)=\int_{-∞}^∞f(x,y)dx$, for the continuous case

## Definition 3.11
>條件分佈

Let $X$ and $Y$ be two random variables, discrete or continuous. 
The **conditional distribution** of the random variable $Y$ given that $X = x$ is
$$
f(y|x) = \frac{f(x, y)} {g(x)} , \text {provided $g(x) > 0$.}
$$
the **conditional distribution** of $X$ given that $Y = y$ is
$$
f(x|y) = \frac{f(x, y)} {h(y)} , \text {provided $h(y) > 0$.}
$$



## Definition 3.12
>獨立事件

Let $X$ and $Y$ be two random variables, discrete or continuous, with joint probability distribution $f(x, y)$ and marginal distributions $g(x)$ and $h(y)$, respectively.
The random variables $X$ and $Y$ are said to be **statistically independent** if and
only if
$$
f(x, y) = g(x)h(y)
$$
for all $(x, y)$ within their range.


## Definition 3.13
Let $X_1,X_2, ... , X_n$ be $n$ random variables, discrete or continuous, with joint probability distribution $f(x_1, x_2, ... , x_n)$ and marginal distribution $f_1 (x_1 ), f_2 (x_2 ), ... , f_n (x_n )$, respectively. The random variables $X_1,X_2, ... , X_n$ are said to be mutually **statistically independent** if and only if
$$
f(x_1, x_2, ... , x_n) = f_1 (x_1 )f_2 (x_2 ) ... f_n (x_n )
$$
for all $(x_1, x_2, ... , x_n)$within their range.

# Chapter 4
## Definition 4.1
>平均、期望值

Let $X$ be a random variable with probability distribution $f(x)$. The **mean**, or **expected value**, of $X$ is
$$
μ = E(X) = 
\left\{ 
  \begin{array}{l}
    \sum\limits_{x}xf(x), & \text{if X is discrete}\\ 
    \int_{-∞}^∞xf(x)\mathrm{d}x, & \text{if X is continuous}
  \end{array}
\right.
$$

## Theorem 4.1
Let $X$ be a random variable with probability distribution $f(x)$. The expected value of the random variable $g(X)$ is
$$
μ_{g(X)} = E[g(X)] = 
\left\{ 
  \begin{array}{l}
    \sum\limits_{x} g(x)f(x), & \text{if X is discrete}\\ 
    \int_{-∞}^∞g(x)f(x)\mathrm{d}x, & \text{if X is continuous}
  \end{array}
\right.
$$

## Definition 4.2
Let $X$ and $Y$ be random variables with joint probability distribution $f(x, y)$. The mean, or expected value, of the random variable $g(X, Y )$ is
$$
μ_{g(X,Y )} = E[g(X, Y )] =
\left\{ 
  \begin{array}{l}
    \sum\limits_{x}\sum\limits_{y}g(x,y)f(x,y), & \text{if X and Y are discrete}\\ 
    \int_{-∞}^∞\int_{-∞}^∞g(x,y)f(x,y)\mathrm{d}x\mathrm{d}y, & \text{if X and Y are continuous}
  \end{array}
\right.
$$

## Definition 4.3
Let $X$ be a random variable with probability distribution $f(x)$ and mean $μ$. The
variance of $X$ is

$$
σ^2=E[(X-μ)^2]=\left\{ 
  \begin{array}{l}
    {\sum\limits_{x}}(x-μ)^2f(x), & \text{if X is discrete}\\ 
    \int_{-∞}^∞(x-μ)^2f(x)\mathrm{d}x, & \text{if X is continuous}
  \end{array}
\right.
$$

The positive square root of the variance, $σ$, is called the **standard deviation** of
$X$.

## Theorem 4.2
The variance of a random variable $X$ is
$$
σ^2=E(X^2)−μ^2.
$$


## Theorem 4.3

Let $X$ be a random variable with probability distribution $f(x)$. The variance of the random variable $g(X)$ is

$$
σ^2_{g(X)}=E\{[g(X)-μ_{g(X)}]^2\}=\left\{ 
  \begin{array}{l}
    {\sum\limits_{x}}[g(x)-μ_{g(X)}]^2f(x), & \text{if X is discrete}\\ 
    \int_{-∞}^∞[g(x)-μ_{g(X)}]^2f(x)\mathrm{d}x, & \text{if X is continuous}
  \end{array}
\right.
$$

## Definition 4.4
Let $X$ and $Y$ be random variables with joint probability distribution $f(x, y)$. The
covariance of $X$ and $Y$ is

$$
σ_{XY}=E[(X−μ_X)(Y−μ_Y)] =
\left\{ 
  \begin{array}{l}
    {\sum\limits_{x}}{\sum\limits_{y}}
(x − μ_X )(y − μ_y )f(x, y), & \text{if X and Y are discrete}\\ 
    \int_{-∞}^∞\int_{-∞}^∞(x − μ_X )(y − μ_y )f(x, y)\mathrm{d}x\mathrm{d}y, & \text{if X and Y are continuous}\\ 
  \end{array}
\right.
$$


## Theorem 4.4
The covariance of two random variables $X$ and $Y$ with means $μ_X$ and $μ_Y$ , respectively, is given by
$$
σ_{XY}=E(XY)−μ_Xμ_Y.
$$

## Definition 4.5
Let $X$ and $Y$ be random variables with covariance $σ_{XY}$ and standard deviations $σ_X$ and $σ_Y$, respectively. The correlation coefficient of $X$ and $Y$ is

$$
\rho_{XY} = \frac{σ_{XY}}{σ_{X}σ_{Y}}.
$$


## Theorem 4.5
If $a$ and $b$ are constants, then
$$
E(aX + b) = aE(X) + b.
$$


## Theorem 4.6
The expected value of the sum or di!erence of two or more functions of a random variable $X$ is the sum or difference of the expected values of the functions. That
is,
$$
E[g(X) ± h(X)] = E[g(X)] ± E[h(X)].
$$


## Theorem 4.7
The expected value of the sum or difference of two or more functions of the random variables $X$ and $Y$ is the sum or difference of the expected values of the functions.
That is,
$$
E[g(X, Y ) ± h(X, Y )] = E[g(X, Y )] ± E[h(X, Y )].
$$

## Theorem 4.8
Let $X$ and $Y$ be two independent random variables. Then
$$
E(XY ) = E(X)E(Y ).
$$

## Theorem 4.9
If $X$ and $Y$ are random variables with joint probability distribution $f(x, y)$ and $a$, $b$, and $c$ are constants, then
$$
σ^2_{aX+bY+c} = a^2σ^2_X + b^2σ^2_Y + 2abσ_{X Y} .
$$

## Theorem 4.10
**(Chebyshev’s Theorem)** The probability that any random variable $X$ will assume a value within $k$ standard deviations of the mean is at least $1 − 1/k^2$. That
is,
$$
P(μ − kσ < X < μ + kσ) \ge 1 −\frac{1}{k^2}.
$$


# Chapter 5

## Binomial Distribution
> 二項式分佈

A Bernoulli trial can result in a success with probability $p$ and a failure with probability $q = 1−p$. Then the probability distribution of the binomial random variable $X$, the number of successes in n independent trials, is
$$
b(x;n,p)=\pmatrix{n\\x}p^xq^{n-x}, ~~~ x=0,1,2,...,n.
$$

## Multinomial Distribution
> 多項分佈

If a given trial can result in the $k$ outcomes $E_1,E_2, . . . , E_k$ with probabilities $p_1, p_2, . . . , p_k$, then the probability distribution of the random variables $X_1,X_2 , . . . , X_k$ , representing the number of occurrences for $E_1,E_2, . . . , E_k$ in $n$ independent trials, is
$$
f(x_1,x_2,...,x_k;p_1,p_2,...,p_k,n)=\pmatrix{n\\x_1,x_2,...,x_k}p_1^{x_1}p_2^{x_2}...p_k^{x_k},
$$

with 

$$
\sum_{i=1}^{k}x_i = n ~~~\text{and}~~~ \sum_{i=1}^{k}p_i = 1.
$$


## Hypergeometric Distribution
> 超幾何分佈

（x：抽到瑕疵的數量，N：總數量，n：抽查數量，k：總瑕疵數量）

The probability distribution of the hypergeometric random variable $X$, the number of successes in a random sample of size $n$ selected from $N$ items of which $k$ are labeled **success** and $N − k$ labeled **failure**, is
$$
h(x; N, n, k) =\frac{\pmatrix{k\\x}\pmatrix{N-k\\n-x}}{\pmatrix{N\\n}}, ~~~ \max\{{0, n − (N − k)}\} \le x \le \min\{{n, k}\}.
$$

## Multivariate Hypergeometric Distribution
> 多變量超幾何分佈

If $N$ items can be partitioned into the $k$ cells $A_1,A_2, ... , A_k$ with $a_1, a_2, ... , a_k$ elements, respectively, then the probability distribution of the random variables $X_1,X_2, ... , X_k$, representing the number of elements selected from $A_1,A_2, ... , A_k$ in a random sample of size $n$, is
$$
f(x_1,x_2,...,x_k;a_1,a_2,...,a_k,N,n)=\frac{\pmatrix{a_1\\x_1}\pmatrix{a_2\\x_2}...\pmatrix{a_k\\x_k}}{\pmatrix{N\\n}},
$$

with $\sum_{i=1}^k x_i=n$ and $\sum_{i=1}^k a_i=N.$

## Negative Binomial Distribution
> 負二項式分佈

If repeated independent trials can result in a success with probability $p$ and a failure with probability $q = 1 − p$, then the probability distribution of the random variable $X$, the number of the trial on which the $k$th success occurs, is
$$
b^*(x; k, p) =\pmatrix{x-1\\k-1}p^k q^{x−k}, ~~~ x= k, k + 1, k + 2, . . . .
$$

## Geometric Distribution
> 幾何分布

If repeated independent trials can result in a success with probability $p$ and a failure with probability $q = 1 − p$, then the probability distribution of the random variable $X$, the number of the trial on which the first success occurs, is
$$
g(x; p) = pq^{x−1}, ~~~ x= 1, 2, 3,...
$$

## Poisson Distribution
The probability distribution of the Poisson random variable $X$, representing the number of outcomes occurring in a given time interval or specified region denoted by $t$, is
$$
p(x;𝜆t) = \frac{e^{−𝜆t}(𝜆t)^x}{x!}, ~~~ x= 0, 1, 2, . . . ,
$$

where $𝜆$ is the average number of outcomes per unit time, distance, area, or volume and $e = 2.71828...$.

## Theorem 5.1
The mean and variance of the binomial distribution $b(x; n, p)$ are
$$
μ = np ~~~ \text{and} ~~~  \sigma^2 = npq.
$$

## Theorem 5.2
The mean and variance of the hypergeometric distribution $h(x; N, n, k)$ are
$$
μ = \frac{nk}{N} ~~~ \text{and} ~~~  \sigma^2 = \frac{N-n}{N-1} \cdot n \cdot \frac{k}{N}(1-\frac{k}{N}).
$$

## Theorem 5.3
The mean and variance of a random variable following the geometric distribution are
$$
μ = \frac{1}{p} ~~~ \text{and} ~~~  \sigma^2 = \frac{1-p}{p^2}.
$$

## Theorem 5.4
Both the mean and the variance of the Poisson distribution $p(x;𝜆t)$ are $𝜆t$.

## Theorem 5.5
Let $X$ be a binomial random variable with probability distribution $b(x; n, p)$. When $n\to \infty$, $p \to 0$, and $np \xrightarrow{n\to \infty} μ$ remains constant,
$$
b(x; n, p) \xrightarrow{n\to \infty} p(x;μ).
$$


# Chapter 6

## Uniform Distribution
The density function of the continuous uniform random variable $X$ on the interval $[~A, B~]$ is
$$
f(x; A,B) = \left\{
    \begin{array}{l}
        \array{
            \frac{1}{B-A}, & A\le x \le B\\
            0, & \text{elsewhere.} 
        }
    \end{array}
\right.
$$

## Normal Distribution
The density of the normal random variable $X$, with mean $μ$ and variance $\sigma^2$, is
$$
n(x; μ, \sigma) =\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2\sigma^2}(x-μ)^2}, ~~~ -\infty \lt x \lt \infty,
$$

where $\pi = 3.14159 . . .$ and $e = 2.71828 . . .$ .

## Normal Approximation to the Binomial Distribution
Let $X$ be a binomial random variable with parameters $n$ and $p$. For large $n$, $X$ has approximately a normal distribution with $μ = np$ and $\sigma^2 = npq = np(1−p)$ and

$$
\eqalign{
P(X \le x) &= \sum_{k=0}^x b(k;n,p) \\
           &\approx \text{area under normal curve to the left of $x + 0.5$} \\
           &= P\pmatrix{Z \le \frac{x+0.5-np}{\sqrt{npq}}},
}
$$

and the approximation will be good if $np$ and $n(1−p)$ are greater than or equal to 5.

## Gamma Distribution
The continuous random variable $X$ has a gamma distribution, with parameters $\alpha$ and $\beta$, if its density function is given by
$$
f(x;\alpha,\beta) = \left\{
    \begin{array}{l}
        \array{
            \frac{1}{\beta^{\alpha}\Gamma(\alpha)}x^{\alpha-1}e^{-x/\beta}, & x \gt 0\\
            0, & \text{elsewhere.} 
        }
    \end{array}
\right.
$$

where $\alpha > 0$ and $\beta > 0$.

## Exponential Distribution
The continuous random variable $X$ has an **exponential distribution**, with parameter $\beta$, if its density function is given by
$$
f(x;\beta) = \left\{
    \begin{array}{l}
        \array{
            \frac{1}{\beta}e^{-x/\beta}, & x \gt 0\\
            0, & \text{elsewhere.} 
        }
    \end{array}
\right.
$$

where $\beta \gt 0.$

## Chi-Squared Distribution
The continuous random variable $X$ has a **chi-squared distribution**, with $v$ **degrees of freedom**, if its density function is given by
$$
f(x;v) = \left\{
    \begin{array}{l}
        \array{
            \frac{1}{2^{v/2}\Gamma(v/2)}x^{v/2-1}e^{-x/2}, & x \gt 0\\
            0, & \text{elsewhere.} 
        }
    \end{array}
\right.
$$

where $v$ is a positive integer.

## Beta Distribution
The continuous random variable $X$ has a **beta distribution** with parameters $\alpha > 0$ and $\beta > 0$ if its density function is given by
$$
f(x) = \left\{
    \begin{array}{l}
        \array{
            \frac{1}{B(\alpha,\beta)}x^{\alpha-1}(1-x)^{\beta-1}, & 0 \lt x \lt 1\\
            0, & \text{elsewhere.} 
        }
    \end{array}
\right.
$$

## Lognormal Distribution
The continuous random variable $X$ has a **lognormal distribution** if the random variable $Y = \ln(X)$ has a normal distribution with mean $μ$ and standard deviation $\sigma$. The resulting density function of $X$ is
$$
f(x;μ,\sigma) = \left\{
    \begin{array}{l}
        \array{
            \frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{1}{2\sigma^2}[\ln(x)-μ]^2}, & x \ge 0\\
            0, & x<0 
        }
    \end{array}
\right.
$$

## Weibull Distribution
The continuous random variable $X$ has a **Weibull distribution**, with parameters $\alpha$ and $\beta$, if its density function is given by
$$
f(x;\alpha,\beta) = \left\{
    \begin{array}{l}
        \array{
            \alpha\beta x^{\beta-1}e^{-\alpha x^{\beta}}, & x > 0\\
            0, & \text{elsewhere.} 
        }
    \end{array}
\right.
$$

where $\alpha > 0$ and $\beta > 0$.

## cdf for Weibull Distribution
The **cumulative distribution function for the Weibull distribution** is given by
$$
F(x)=1-e^{-\alpha x^{\beta}}, ~~~ \text{for} ~ x \ge 0,
$$

where $\alpha > 0$ and $\beta > 0$.

## Failure Rate for Weibull Distribution
The **failure rate** at time t for the Weibull distribution is given by
$$
Z(t)=\alpha\beta t^{\beta-1}, ~~~ t>0.
$$

## Theorem 6.1
The mean and variance of the uniform distribution are
$$
μ=\frac{A+B}{2} ~~~ \text{and} ~~~  \sigma^2 = \frac{(B-A)^2}{12}.
$$

## Theorem 6.2
The mean and variance of $n(x; μ,\sigma)$ are $μ$ and $\sigma^2$, respectively. Hence, the standard deviation is $\sigma$.

## Definition 6.1
The distribution of a normal random variable with mean 0 and variance 1 is called a **standard normal distribution**.

## Theorem 6.3
If $X$ is a binomial random variable with mean $μ = np$ and variance $\sigma^2 = npq$, then the limiting form of the distribution of
$$
Z=\frac{X-np}{\sqrt{npq}},
$$
as $n\to \infty$, is the standard normal distribution $n(z; 0, 1)$.

## Definition 6.2
The gamma function is defined by
$$
\Gamma(\alpha) = \int_0^\infty x^{\alpha-1}e^{-x} \mathrm{d} x , ~~~ \text{for} ~~~ \alpha >0,
$$

## Theorem 6.4
The mean and variance of the gamma distribution are
$$
μ = \alpha\beta ~~~ \text{and} ~~~  \sigma^2 = \alpha\beta^2 .
$$

## Theorem 6.5
The mean and variance of the chi-squared distribution are
$$
μ = v ~~~ \text{and} ~~~ \sigma^2 = 2v.
$$

## Definition 6.3
A beta function is defined by
$$
B(\alpha,\beta) = \int_0^1 x^{\alpha-1}(1-x)^{\beta-1} \mathrm{d} x = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}, ~~~ \text{for} ~~~ \alpha,\beta >0,
$$
where $\Gamma(\alpha)$ is the gamma function.

## Theorem 6.6
The mean and variance of a beta distribution with parameters $\alpha$ and $\beta$ are
$$
μ = \frac{\alpha}{\alpha+\beta} ~~~ \text{and} ~~~ \sigma^2 = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)},
$$
respectively.

## Theorem 6.7
The mean and variance of the lognormal distribution are
$$
μ = e^{μ+\sigma^2 /2} ~~~ \text{and} ~~~ \sigma^2 = e^{2μ+\sigma^2} (e^{\sigma^2} − 1).
$$

## Theorem 6.8
The mean and variance of the Weibull distribution are
$$
μ = \alpha^{-1/\beta} \Gamma\left(1 + \frac{1}{\beta}\right) \quad \text{and} \quad \sigma^2 = \alpha^{-2/\beta} \left\{ \Gamma\left(1 + \frac{2}{\beta}\right) - \left[ \Gamma\left(1 + \frac{1}{\beta}\right) \right]^2 \right\}.
$$

# Chapter 8

## Definition 8.1
A **population** consists of the totality of the observations with which we are concerned.

## Definition 8.2
A **sample** is a subset of a population.

## Definition 8.3
Let $X_1,X_2, . . . , X_n$ be n independent random variables, each having the same probability distribution $f(x)$. Define $X_1,X_2, . . . , X_n$ to be a **random sample** of size $n$ from the population $f(x)$ and write its joint probability distribution as
$$
f(x_1, x_2, . . . , x_n) = f(x_1)f(x_2) · · · f(x_n).
$$

## Definition 8.4
Any function of the random variables constituting a random sample is called a **statistic**.

## Theorem 8.1
If $S^2$ is the variance of a random sample of size $n$, we may write
$$
S^2 = \frac{1}{n(n-1)}= \left[ n\sum_{i=1}^nX_i^2 - \left(\sum_{i=1}^nX_i\right)^2 \right].
$$

## Definition 8.5
The probability distribution of a statistic is called a **sampling distribution**.

## Theorem 8.2
**Central Limit Theorem:** If $\bar X$ is the mean of a random sample of size $n$ taken from a population with mean $μ$ and finite variance $\sigma^2$, then the limiting form of the distribution of
$$
Z = \frac{\bar X - μ}{\sigma / \sqrt{n}},
$$

as $n\to \infty$, is the standard normal distribution $n(z; 0, 1)$.

## Theorem 8.3
If independent samples of size $n_1$ and $n_2$ are drawn at random from two populations, discrete or continuous, with means $μ_1$ and $μ_2$ and variances $\sigma^2_1$ and $\sigma_2^2$, respectively, then the sampling distribution of the di!erences of means, $\bar X_1$ − $\bar X_2$, is approximately normally distributed with mean and variance given by
$$
μ_{\bar X_1 − \bar X_2} = μ_1 - μ_2 ~~~\text{and}~~~ \sigma^2_{\bar X_1 − \bar X_2} = \frac{\sigma^2_1}{n_1} + \frac{\sigma^2_2}{n_2}.
$$

Hence, 
$$
Z = \frac{(\bar X_1 − \bar X_2)-(μ_1 - μ_2)}{\sqrt{(\sigma^2_1/n_1)+(\sigma^2_2/n_2)}}
$$

is approximately a standard normal variable.

## Theorem 8.4
If $S^2$ is the variance of a random sample of size n taken from a normal population having the variance $\sigma^2$, then the statistic
$$
X^2 = \frac{(n-1)S^2}{\sigma^2} = \sum_{i=1}^n\frac{(X_i-\bar X)^2}{\sigma^2}
$$

has a chi-squared distribution with $v = n − 1$ degrees of freedom.

## Theorem 8.5
Let $Z$ be a standard normal random variable and $V$ a chi-squared random variable with $v$ degrees of freedom. If $Z$ and $V$ are independent, then the distribution of the random variable $T$, where
$$
T = \frac{Z}{\sqrt{V/v}},
$$

is given by the density function
$$
h(t) = \frac{\Gamma [(v+1)/2]}{\Gamma (v/2)\sqrt{\pi v}}\left(1+\frac{t^2}{v}\right)^{-(v+1)/2}, ~~~ -\infty < t < \infty.
$$

This is known as the **$t$-distribution** with $v$ degrees of freedom.

## Theorem 8.6
Let $U$ and $V$ be two independent random variables having chi-squared distributions with $v_1$ and $v_2$ degrees of freedom, respectively. Then the distribution of the random variable $F = \frac{U/v_1}{V/v_2}$ is given by the density function
$$
h(t) = \left\{
    \begin{array}{l}
        \array{
            \frac{\Gamma [(v_1+v_2)/2](v_1/v_2)^{v_1/2}}{\Gamma (v_1/2)\Gamma (v_2/2)}\frac{f^{(v_1/2)-1}}{(1+v_1f/{v_2})^{(v_1+v_2)/2}}, & f>0\\
            0, & f \le 0 
        }
    \end{array}
\right.
$$

This is known as the **$F$-distribution** with $v_1$ and $v_2$ degrees of freedom (d.f.).

## Theorem 8.7
Writing $f_{\alpha} (v_1, v_2)$ for $f_{\alpha}$ with $v_1$ and $v_2$ degrees of freedom, we obtain
$$
f_{1-\alpha}(v_1,v_2) = \frac{1}{f_{\alpha} (v_2, v_1)}.
$$

## Theorem 8.8
If $S^2_1$ and $S^2_2$ are the variances of independent random samples of size $n_1$ and $n_2$ taken from normal populations with variances $\sigma^2_1$ and $\sigma^2_2$, respectively, then
$$
F = \frac{S^2_1/\sigma^2_1}{S^2_2/\sigma^2_2} = \frac{\sigma^2_2S^2_1}{\sigma^2_1S^2_2}
$$

has an $F$-distribution with $v_1 = n_1 − 1$ and $v_2 = n_2 − 1$ degrees of freedom.

## Definition 8.6
A **quantile** of a sample, $q(f)$, is a value for which a specified fraction $f$ of the data values is less than or equal to $q(f)$.

## Definition 8.7
The **normal quantile-quantile plot** is a plot of $y_{(i)}$ (ordered observations) against $q_{0,1}(f_i)$, where $f_i = \frac{i-\frac{3}{8}}{n+\frac{1}{4}}$.