After scoring a touchdown, a football team may elect to attempt a two-point conversion, by running or passing the ball into the end zone. If successful, the team scores two points. For a certain football team, the probability that this play is successful is 0.40.

a. Let X = 1 if successful, X = 0 if not. Find the mean and variance of X.

b. If the conversion is successful, the team scores 2 points; if not the team scores 0 points. Let Y be the number of points scored. Does Y have a Bernoulli distribution? If so, find the success probability. If not, explain why not.

c. Find the mean and variance of Y.

Answer:

Step 1 of 3:

(a)

In this question, we are asked to find the mean and variance of .

Where is random variable of team scoring successfully two points in the football game.

For a certain football team, the probability that this play is successful is .

Let if successful, if not.

If the random event results in success, then . Otherwise . It follows that is a discrete random variable, with probability mass function (PMF) defined by

for any value of x other than 0 or 1

The random variable is said to have the Bernoulli distribution with parameter .

And the notation is ∼ Bernoulli().

Mean) of a Bernoulli random variable

………….(1)

Where is probability of success of any event which is given in the question.

Substitute the value of into the equation (1)

Variance of a Bernoulli random variable

………….(2)

Substitute the value of into the equation (2)

Hence mean and variance of is and respectively.