Random Variables, CDF, PMF, and Expectation

Cumulative Distribution Function (CDF)

• The Cumulative Distribution Function (CDF), denoted as $F<em>X(x)$, represents the probability that a random variable $X$ takes a value less than or equal to a specific real value $x$. Mathematically, this is expressed as: $F</em>X(x) = P(X \le x)$.

  • Here, x is a small x representing a real value, while X (or Y) is a big X representing the random variable.

  • The definition strictly includes the equal sign, meaning if $X$ can take the value x, that probability is included.

• Example 4.1 (Y takes 0 or 1):

  • If x < 0, then $P(Y \le x) = 0$, so $F_Y(x) = 0$. This is because there are no possible values for Y less than 0 in this region.

  • If 0 \le x < 1, then $P(Y \le x) = P(Y=0) = 1/2$ (assuming Y=0 has probability 1/2). The region covers 0 but not 1 or greater.

  • If $x \ge 1$, then $P(Y \le x) = P(Y=0) + P(Y=1) = 1/2 + 1/2 = 1$. The region covers both 0 and 1 (and all values greater).

• Plotting the CDF:

  • The CDF is a step function for discrete random variables.

  • It starts at 0 for values less than the smallest possible outcome and increases in steps at each possible outcome.

  • For a discrete random variable, the CDF may not be continuous; it can have