The following table is a valid probability distribution, for a random variable X, where the individual probabilities are unknown. $$\begin{array} { | c | c | }
\hline x & P [ X ] \
\hline - 3 & \
\hline - 2 & \
\hline 0 & \
\hline 1 & \
\hline 2 & \
\hline
\end{array}$$ Which of the following statements is correct ?
A) The expected value of $X$ can only be 0 .
B) The expected value of $X$ could be 3 .
C) The expected value of $X$ could be $0.5$
D) None of these choices are correct

Question

The following table is a valid probability distribution, for a random variable X, where the individual probabilities are unknown. $$\begin{array} { | c | c | } 
\hline x & P [ X ] \
\hline - 3 & \
\hline - 2 & \
\hline 0 & \
\hline 1 & \
\hline 2 & \
\hline
\end{array}$$ Which of the following statements is correct ? 
A) The expected value of $X$ can only be 0 .
B) The expected value of $X$ could be 3 .
C) The expected value of $X$ could be $0.5$
D) None of these choices are correct

Quizplus · Accepted Answer

In order for the expected value of X to be 0, the sum of the products of each possible value of X and their corresponding probabilities must equal 0. However, since we do not have the individual probabilities, we cannot determine if this is possible. Similarly, in order for the expected value to be 3, the sum of the products of each possible value of X and their corresponding probabilities must equal 3. This is not possible since the highest possible value of X is 2. However, it is possible for the expected value of X to be 0.5 by assigning probabilities such that P[-3] = P[-2] = P[2] = 0.1, P[0] = 0.5, and P[1] = 0.2. Then, the expected value of X is calculated as (-3)(0.1) + (-2)(0.1) + (0)(0.5) + (1)(0.2) + (2)(0.1) = 0.5. Therefore, choice C is correct.

The Following Table Is a Valid Probability Distribution, for a Random