Question

In college basketball, when a player is fouled while not in the act of shooting and the opposing team is “in the penalty,” the player is awarded a “1 and 1.” In the 1 and 1, the player is awarded one free throw, and if that free throw goes in the player is awarded a second free throw.

a) Find the PMF of Y, the number of points scored in a 1 and 1 given that any free throw goes in with probability p, independent of any other free throw.

b) Find and sketch the CDF of Y, the number of points scored in a 1 and 1 for p = 1/4, p = 1/2, and p = 3/4.

Answer 1

a)

Let X: at that point score in a 1 and 1 .

at that point

P(X=0, i.e scoring 0 ) = likelihood that the player misses first free hit it self = 1-p (given)

P(x=1) = likelihood that the player scores an objective in first free throw and misses the second =

(P)*(1-p)

likewise P(x=n) = likelihood that the player scores a goal in all first n tosses and misses the n+1 st one. =pⁿ (1-p)

The probabilities are the PMF s for the inquiry and the general formula for x=n is likewise above.

b)

The cumulative distribution function (CDF) of random variable X is defined as

FX(x)=P(X≤x), for all x∈R.