GATE CS 2012 Q.21 (CS Q.21): Probability and Statistics

Consider a random variable $X$ that takes values $+1$ and $-1$ with probability $0.5$ each. The values of the cumulative distribution function $F(x)$ at $x = -1$ and $+1$ are

0 and 0.5

0 and 1

0.5 and 1

0.25 and 0.75

Official answer and solution

Official answer: C
From: Official GATE 2012 answer key

Solution

Apply the definition of the Cumulative Distribution Function (CDF) for a discrete random variable.

For any random variable $X$ , the cumulative distribution function is defined as:

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

The random variable $X$ takes two discrete values $+1$ and $-1$ with probabilities:

P(X = -1) = 0.5, \quad P(X = +1) = 0.5

Evaluating $F(x)$ at $x = -1$ :

F(-1) = P(X \le -1) = P(X = -1) = 0.5

Evaluating $F(x)$ at $x = +1$ :

F(+1) = P(X \le +1) = P(X = -1) + P(X = +1) = 0.5 + 0.5 = 1

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

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2012 Computer Science and Information Technology (CS)
Question: 21 of 65, printed as CS Q.21
Section: Computer Science and Information Technology

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