Functions

Functions of Discrete Random Variables

Functions of random variables are also random variables.

If $Y=g(x)$ then we have:

$$p_y(y)=\sum _{\{x|g(x)=y\}}{p}_x(x)$$

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable is the function:

$$F(x)=P(X\le x)$$

The CDF gives us the probability that the variable takes a value less than or equal to $x$.

A property of CDF’s:

1. $0\le F(X)\le 1$ for all $x$.
2. $x_0<x_1\to F_X(x_0)\le F_X(x_1)$

Example: Obtaining the CDF from the PMF

Suppose the range of a discrete random variable is $\{0,1,2,3,4\}$ and its probability mass function is $P(X=x)=\frac{x}{10}$

For $x<1$, $F(x)=\sum_

Functions of Continuous Variables

Probability Density Function (PDF)

The probability is given by the area under the curve.