Question

Question 1:

Jaycar, an electronics store sponsors the Canterbury Bulldogs, a team that competes in the NRL. An individual store owner is looking to promote the sale of their 160W solar panels and is planning to offer a discount on the price of the panels from Monday to Friday based on the number of home games won in the past weekend of NRL. The owner of the store has asked us to do some analysis of the promotion assuming the following:

(a) The number of wins each weekend out of eight games is a binomial random variable with a probability of the home team winning being equal to p.

(b) The number of solar panels normally sold from Monday to Friday is a Poisson random variable with mean units sold equal to λ.

(c) The number of wins in a weekend for home teams in the NRL and the number of solar panels sold from Monday to Friday are independent.

Question 1 A):

If we let X represent the number of home team wins out of eight games in a weekend of NRL, what will the mean and variance of X be in terms of p.

Question 1 B):

If we let Y represent the number of 160W solar panels sold from Monday to Fridays, what will the mean and variance of Y be in terms of λ.

Question 1C):

The “cost” of the promotion will be a new random variable that is a function of:

the number of home wins over the weekend;

the number of solar panels that will be sold; and the discount offered per game. If we let the cost be a new random variable such that Z = a*X*Y , find the mean and variance of Z. Hint: The variables are assumed to be independent.

The variance of the product of independent variables is given by

Var( XY ) = (E( X))^2*Var( Y ) + (E( Y ))^2*Var(X) + Var(X)*Var( Y ),

and Var(c*X) = c^2*Var(X).

Answer 1

QUESTION 1A):

   Given that the number of wins each weekend out of eight games is a binomial random variable with a probability of the home team winning being equal to p. And let X represent the number of home team wins out of eight games in a weekend of NRL.

TO PROVE:

   The mean and variance of X be in terms of p.

PROOF:

   The probability density function of binomial random variable X is given by,

  

where p is the probability of home team winning and q is the probability of losing.

Now the mean of binomial random variable X is given by,

  

  ​ since for x=0,

By using factors of binomial coefficient, that is for all and :

  

Thus we have

​

​

{By taking out np and by using n-x=(n-1)-(x-1)}

   {By putting m=n-1 and j=x-1}

Now sum of all binomial terms =sum of all probabilities=1

Thus we have

=

Since the value of n=8 {that is there are totally eight games, the mean of X is 8p.

VARIANCE OF BINOMIAL RANDOM VARIABLE X:

QUESTION 1B):

   Given that the number of solar panels normally sold from Monday to Friday is a Poisson random variable with mean units sold equal to . And let Y represent the number of 160W solar panels sold from Monday to Fridays,

TO PROVE:

   The mean and variance of Y be in terms of .

PROOF:

QUESTION 1C):

Let the cost be a new random variable such that Z = a*X*Y .

TO FIND:

   The mean and variance of Z

PROOF: