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# System prepended metadata

title: Week 2 Lecture 2

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# Week 2 Lecture 2
## Random Variable
For a given sample space &Omega; of some experiment, a **random variable** is any rule that associates a real number with each outcome in &Omega;.
A random variable is a **real-valued** function (its range is within the set of real numbers)  whose domain is the sample space.
![](https://i.imgur.com/qlGO9dH.png)

## Discrete and Continuous RV
A random variable is called **discrete** if X can take only a finite number *k* of different values x~1~,...,x~k~, or, at most, an infinite sequence of different values x~1~,x~2~,x~3~,...

A random variable X is called continuous if the following conditions hold:
* The set of all possible values for X is either a single interval on the real line, or a union of disjoint intervals on the real line.
* No possible value has positive probability: $Pr(X=x) = 0$ for any possible $x$

:::info
P(X=x) means probability of random variable X to take the value x.
:::

### Examples
#### Discrete RV
![](https://i.imgur.com/dD13TlX.png)

#### Continuous RV
![](https://i.imgur.com/Xi8EAzD.png)
![](https://i.imgur.com/nGwNH37.png)

#### Question
![](https://i.imgur.com/vvxbhFX.png)
There is a discrete component, which is the element 2, and a continuous component, which is [0,1]

## Probability Distribution
Let X be a random variable defined on the sample space &Omega;.
Thus, X(&omega;) is a real number for every outcome &omega; &isin; &Omega;.
Let C be a subset of the real line. We write {X &isin; C} to mean the set {&omega; &isin; &Omega;: X(&omega;) &isin; C} of all outcomes whose X-value is in C.
![](https://i.imgur.com/8jacEkP.png)

:::info
An event is a subset of outcomes contained in &Omega;. Hence, {X &isin; C} is an event.
:::

Every event is assigned a probability, so it makes sense to consider the probability of {X &isin; C}:
$Pr(X \in C ) = Pr(\{\omega\in\Omega:X(\omega)\in C\})$

The probability distribution of X is the collection of all probabilities of the form Pr(X&isin;C), for all sets C of real numbers.

In other word, for any set C of real numbers, the probability distribution of X gives the probability Pr(X&isin;C) of how likely the random variable X takes on values in C.
## Probability Mass Distribution (pmf)
Let X be a discrete RV defined on the sample space &Omega;.
The pmf of X is a function p(x) defined on every real number, such that
$$
\begin{eqnarray*}
p(x) &=& Pr(X=x) \\
     &=& Pr({\omega\in\Omega:X(\omega)=x})
\end{eqnarray*}
$$
The pmf of X completely describes the distribution of X, since
$$
Pr(X\in C)=\sum_{x\in C}p(x).
$$
### Example
![](https://i.imgur.com/pr1eBpM.png)
### Bernoulli Distribution
An RV X is called a Bernoulli RV if it takes only 2 values: 0 and 1.
The pmf of X is given by
$Pr(X=1)=p$, 
$Pr(X=0)=1-p$, for some 0 &le; p &le; 1.
We say that X is the Bernoulli R.V. with parameter p.
![](https://i.imgur.com/2DhjLC5.png)

### Bernoulli Process and Binomial RV
Consider an experiment (a Bernoulli process) with n repeated trials, such that:
* The trials are independent
* Each trial has only 2 outcomes: 1(success) and 0 (failure)
* The success rate of the trials is the same (denoted by some probability p).

The pmf of binomial RV is given by:
![](https://i.imgur.com/T2KLpSh.png)
where n is the number of trials, and p is the success rate of each trial.
![](https://i.imgur.com/pWUfdhY.png)

### Examples
![](https://i.imgur.com/dYDoABn.png)
![](https://i.imgur.com/c1dlhbo.png)
![](https://i.imgur.com/4dA85PX.png)

## Probability Density Distribution (pdf)
The pdf of a continuous RV X is a function f(x) such that for any two numbers a and b with a &le; b,
$$
P(a\le X \le b)= \int_a^b f(x) dx
$$
The pdf completely describes the distribution of X.

![](https://i.imgur.com/KD6XvFF.png)

This graph is also known as the density curve.

For a function f(x) to be a valid pdf of some continuous RV, it must satisfy the following conditions:
1. $f(x) \ge 0$ for al x. (Density cannot be negative)
2. $\int_{-\infty}^{\infty}f(x) dx= 1$ (Area under the curve of f(x) is 1)

![](https://i.imgur.com/SmSlxRi.png)

### Uniform Distribution
![](https://i.imgur.com/yrpRVH9.png)

### Exponential Distribution
![](https://i.imgur.com/szQfgTs.png)

### Examples
![](https://i.imgur.com/o5yxxC7.png)
![](https://i.imgur.com/2o6pYoC.png)
![](https://i.imgur.com/DmjYiGK.png)
<hr/>

![](https://i.imgur.com/rsGhuzR.png)
<hr/>
![](https://i.imgur.com/v838Ki5.png)
