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title: Stat06
tags: [Stat]

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title: Stat06
tags: Stat
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1. Ch5: Binomial, Poisson


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### III. Normal random variable :star::star::star::star::star:
1. Ch6


| | Discrete RV | Continous RV| 
| --- |-----|----|
|probability distribution | $p(x)$: probablity mass function (pmf) |  $f(x)$: probability density function (pdf)|
| Easy to understand its idea | :+1:  $p(k)=P(X=k)$| $P(a<X<b) = \int_a^b f(x) dx$| 
| Easy to calculate its probability| | :+1: through calculus| 


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#### Normal distribution



## 6.1


### $X\sim U(a, b)$ uniform random variable (a<b).

1. If $X\sim U(a, b)$, then it has a pdf
$$f(x) = \frac{1}{b-a},$$
for $a<x<b$. 


### Exercise


#### 6.1.18 Coating Thickness

The thickness in microns ($\mu$) of a protective coating applied to a conductor designed to work in corrosize conditions is uniformly distributed on the interval from 25 to 50. 

a. What is the probability that the thickness of the coating is grater than 45 microns?
b. What is the probability that the thickness of the coating is between 35 and 45 microns?
c. What is the probability that the thickiness of the coating is less than 40 micros? 


### Exponential distribution, $X\sim Exp(\lambda)$

1. Suppose $X\sim Exp(\lambda)$. Then, $X$ has pdf
$$f(x) = \lambda e^{-\lambda x},$$
for $x>0$. 
2. $P(X\leq x) =1-e^{-\lambda x}$ for $x>0$.
3. $P(X>x) = 1-P(X\leq x) = e^{-\lambda x}$ for $x>0$. 
4. $lambda$ is called the rate parameter. 
5. $X$ has the mean $\mu=\frac{1}{\lambda}$, variance = ? 


#### 6.1.9-14


#### 6.1.19 Battery Life

The length of life (in days) of an alkaline battery has an exponential distribution with an average life of 1 year, so that $\lambda = 1/365$. 
a. What is the probability that an alkaline battery will fail before 180 days?
b. What is the probability that an alkaline battery will last beyond 1 year?
c. If a device requires two batteries, what is the probability that both batteries last beyond 1 year? 








## 6.2 

#### p7.

If $X\sim N(\mu,\sigma^2)$, then $\frac{X-\mu}{\sigma}\sim N(0, 1)$. 

1. Find the probability
2. Given $p$, find $z_0$, so that $P(Z\leq z_0)=p$.




### 1. $Z \sim N(0,1)$ 
#### 1. $P(a<Z<b)$

Note the table only gives left-tailed probability. But, you should be able to find:
1. 左尾機率: $P(Z\leq 1.36)$
2. 右尾機率: $P(Z>1.36)$
3. 區間機率: $P(-1.20\leq Z\leq 1.36)$


#### 2. 


### 2. $X\sim N(\mu, \sigma^2)$
$X \sim N(\mu,\sigma^2)$


Note: 
$$\frac{X-\mu}{\sigma}\sim N(0,1).$$

#### 
1. 左尾機率: $P(X\leq a)=P(\frac{X-\mu}{\sigma} = \frac{a-\mu}{\sigma}) = P(Z\leq \frac{a-\mu}{\sigma})$
2. 右尾機率:  $P(X>a) =P(\frac{X-\mu}{\sigma} > \frac{a-\mu}{\sigma})=P(Z>\frac{a-\mu}{\sigma})$
3. 區間機率: $P(a<X<b)=P(\frac{a-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{b-\mu}{\sigma})=P(\frac{a-\mu}{\sigma}<Z<\frac{b-\mu}{\sigma})$



#### 2. Given $p$, find $x$? 