Chapter 1.2.2 Special Continuous Random Variables

Uniform Random Variable

A uniform random $X$ over the interval $(a,b)$ satisfies the property that for any interval inside of $(a,b)$ , the probability that $X$ is in that interval depends only on the length of the interval. The pdf is given by

Example

A random number is chosen between 0 and 10. What is the probability that it is bigger than 7, given that it is bigger than 6?

Exponential Random Variable

An exponential random variable $X$ with rate $λ$ has pdf

Exponential rv’s are useful for, among other things, modeling the waiting time until the first occurrence in a Poisson process. So, the waiting time until an electronic component fails, or until a customer enters a store could be modeled by exponential random variables.

The mean of an exponential random variable is $μ=1/λ$ and the variance is $σ2=1/λ2$ .

Here are some plots of pdf’s with various rates.

Normal Random Variable

A normal random variable with mean $μ$ and standard deviation $σ$ has pdf given by

The mean of a normal rv is $μ$ and the standard deviation is $σ$ .

We will see many uses for normal random variables throughout the book, but for now let’s just compute some probabilities when $X\simNormal(μ=2,σ=2).$

$P(X\leq4)$ = pnorm(4,mean = 2,sd = 4)
$P(0\leqX\leq2)$ = pnorm(2,2,4) - pnorm(0,2,4).
Find the value of $q$ such that $P(X\leqq) = .75$ . $q$ = qnorm(.75,2,4). We get about 4.7; we can check it with pnorm(4.7,2,4).