Question

Suppose a baseball player had 231 hits in a season. In the given probability​ distribution, the random variable X represents the number of hits the player obtained in a game. x 0 1 2 3 4 5 ​P(x) 0.1916 0.4193 0.2462 0.1047 0.0118 0.0264 ​(a) Compute and interpret the mean of the random variable X. mu Subscript xequals nothing ​(Round to one decimal place as​ needed.) Which of the following interpretation of the mean is​ correct? A. The observed value of the random variable will almost always be equal to the mean of the random variable. B. As the number of trials n​ decreases, the mean of the observations will approach the mean of the random variable. C. The observed value of the random variable will almost always be less than the mean of the random variable. D. As the number of trials n​ increases, the mean of the observations will approach the mean of the random variable. ​(b) Compute the standard deviation of the random variable X. sigma Subscript xequals nothing ​(Round to one decimal place as​ needed.)

Answer 1

X	0	1	2	3	4	5
P(x)	0.1916	0.4193	0.2462	0.1047	0.0118	0.0264

(a)

Mean = E(x) = x * P(x)

= 0 * 0.1916 + 1 * 0.4193 + 2 * 0.2462 + 3 * 0.1047 + 4 * 0.0118 + 5 * 0.0264

= 1.405

= 1.4

Interpretation of the mean is​

D. As the number of trials n​ increases, the mean of the observations will approach the mean of the random variable.

(b)

E(x²) = x² * P(x)

= 0² * 0.1916 + 1² * 0.4193 + 2² * 0.2462 + 3² * 0.1047 + 4² * 0.0118 + 5² * 0.0264

= 3.1952

Variance = E(x²) - [E(x)]² = 3.1952 - 1.405²  = 1.221175

Standard deviation of the random variable X, = 1.1