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.
Step 1 of 3:
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
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
Substitute the value of into the equation (2)
Hence mean and variance of is and respectively.