Discrete Random Variables

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# Discrete Random Variables
### Dr. Dogucu

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 Copyright &copy; <a href="https://mdogucu.ics.uci.edu">Dr. Mine Dogucu</a>. <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a></div>

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## Discrete Random Variables

A numeric outcome of a random process is called a random variable.

A **discrete random** variable has a countable number of possible outcomes.

We will use capital letters towards the end of the alphabet such as `$X, Y, Z$` to represent random variables.

We will use the corresponding lower case letters `$x, y, z$` to represent observed outcomes.

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## Discrete Random Variable

Random process: 3 flips of a single coin

Sample Space: `$\{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\}$`

Let X be a discrete random variable that represents the total number of heads in 3 coin flips.

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## Discrete Random Variable

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`$P(X = 2) =f(2)=\frac{3}{8} = 0.375$`

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## Probability Mass Function

For discrete random variable, `$X$` the distribution of all possible values of `$x$` can be shown with a probability mass function (pmf).

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## Probability Mass Function (pmf)

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`$P(X = 2) =f(2)=\frac{3}{8} = 0.375$`

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## Cumulative Distribution Function (cdf)

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`$P(X \leq 2) =F(2)=\frac{7}{8} = 0.875$`

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## Cumulative Distribution Function

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## Expected Value

If we repeat this random process of 3 coin flips many times, how many number of heads would we expect to see on average?

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`$E(X) = \sum_{S} x f(x)$`

`$E(X) = (0\cdot 0.125) + (1\cdot 0.375) + (2\cdot 0.375) + (3\cdot 0.125)$`

`$E(X) = \mu = 1.5$`

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## Variance

Variance is the average squared deviation from the average.

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## Variance

~~Variance is the average squared deviation from the average.~~

Variance is the expected squared deviation from the expected value.

`$$Var(X)= E[(X-E(X))^2]$$`
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`$$Var(X)= E[(X-\mu)^2]$$`
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`$$Var(X)= E(X^2) - [E(X)]^2$$`
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## Variance

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`$Var(X)= E(X^2) - [E(X)]^2$`

`$E(X^2) = \sum_{S} x^2 f(x)$`

`$E(X^2) = (0^2\cdot 0.125) + (1^2\cdot 0.375) + (2^2\cdot 0.375) + (3^2\cdot 0.125) = 3$`

`$Var(X)= 3 - 1.5^2 = 0.75$`

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## Linear Combination of Random Variables

Let `$X$` and `$Y$` represent random variables and `$a$` and `$b$` represent constants. Then

`$E(aX +bY) = a\cdot E(X) + b \cdot E(Y),$`

and, if `$X$` and `$Y$` are independent,

`$Var(aX +bY) = a^2\cdot Var(X) + b^2 \cdot Var(Y)$`