GATE CS 2012 Q21 — Probability: Random Variables and Distributions

The cumulative distribution function (CDF) of a random variable $X$ is defined as:

We are given that the random variable $X$ takes values $-1$ and $+1$ with equal probability:

Let us calculate the CDF values at $x = -1$ and $x = +1$:

1. At $x = -1$: $$F(-1) = P(X \le -1)$$ Since $-1$ is the smallest value $X$ can take, the only outcome satisfying $X \le -1$ is $X = -1$. $$F(-1) = P(X = -1) = 0.5$$
2. At $x = +1$: $$F(1) = P(X \le 1)$$ Both possible values of $X$ (i.e., $-1$ and $+1$) satisfy $X \le 1$. $$F(1) = P(X = -1) + P(X = 1) = 0.5 + 0.5 = 1$$

Thus, the values of $F(x)$ at $x = -1$ and $x = +1$ are $0.5$ and $1$ respectively.