Consider a discrete random variable Y, which represents the number of “heads” that will be obtained in three flips of a fair coin. It has PF

a. Calculate the mean of Y.

b. Calculate the variance of Y.

c. Calculate the standard deviation of Y.

d. Calculate the skewness of Y.

e. Calculate the kurtosis of Y.

f. Calculate a .10 quantile of Y.

g. Calculate a .875 quantile of Y.

a. By [3.3],

b. By [s1] and [3.7],

c. The standard deviation of a random variable is simply the square root of its variance:

d. By [3.8],

we obtain from item (c). Applying [3.5] with function f(y) = (y – )³, we obtain:

Accordingly,

e. By [3.8],

[3.11]

we obtain from item (c). Applying [3.5] with function f(y) = (y – )⁴, we obtain:

Accordingly,

f. To determine the .10-quantile of Y, we construct the CDF of Y from its PF , which is given by [3.12]. We obtain:

A .10-quantile is any value y for which (y) = .10. In [s16], we see that there is no such value. Accordingly, a .10-quantile of Y does not exist.

g. A .8758quantile is any value y for which (y) = .875. In [s16], we see that all values y in the interval [2, 3) satisfy this condition, so they are all .875 quantiles of Y.