Question

In the probability distribution to the? right, the random variable X represents the number of hits a baseball player obtained in a game over the course of a season.

x P(x)

0 0.1665

1 0.3356

2 0.2873

3 0.1481

4 0.0366

5 0.0259

(1) Compute and interpret the mean of the random variable X.

(2) Which of the following interpretations of the mean is? correct?

A. In any number of? games, one would expect the mean number of hits per game to be the mean of the random variable.

B. The observed number of hits per game will be equal to the mean number of hits per game for most games.

C. The observed number of hits per game will be less than the mean number of hits per game for most games.

D. Over the course of many? games, one would expect the mean number of hits per game to be the mean of the random variable. ?

(3) Compute the standard deviation of the random variable X.

(4) ?What is the probability that in a randomly selected? game, the player got 2? hits?

(5) What is the probability that in a randomly selected? game, the player got more than 1? hit?

Answer 1

1) mean(E(X)) = 0 * 0.1665 + 1 * 0.3356 + 2 * 0.2873 + 3 * 0.1481 + 4 * 0.0366 + 5 * 0.0259 = 1.6304

2) Option - D

3) E(X^2) = 0^2 * 0.1665 + 1^2 * 0.3356 + 2^2 * 0.2873 + 3^2 * 0.1481 + 4^2 * 0.0366 + 5^2 * 0.0259 = 4.0508

Variance(V(X)) = E(X^2) - (E(X))^2

                        = 4.0508 - (1.6304)^2

                        = 1.393

Standard deviation = sqrt(1.393) = 1.18

4) P(X = 2) = 0.2873

5) P(X > 1) = 1 - P(X < 1)

                   = 1 - (P(X = 0) + P(X = 1))

                   = 1 - (0.1665 + 0.3356)

                   = 1 - 0.5021

                   = 0.4979