G Consider the experiment of a single roll of an honest die and a single toss of 3 fair coins. Let X be the value on the die and let Y be the number of heads obtained on the coins. Find the joint probability function of X and Y

The probability function of X and Y is

With k in {1,2,3,4,5,6}

Step-by-step explanation:

We can naturally assume that X and Y are independent. Because of that, P(X=a, Y=b) = P(X=a) * P(Y=b) for any a, b.

Note that, since the die is honest, then P(X=k) = 1/6 for any k in {1,2,3,4,5,6}. We can conclude as a consequence that P(X=k, Y=l) = P(Y=l)/6 for any k in {1,2,3,4,5,6}.

Y has a binomial distribution, with parameters n = 3, p = 1/2. Y has range {0,1,2,3}. Lets compute the probability mass function of Y:

Thus, we can conclude that the joint probability function is given by the following formula

For any k in {0,1,2,3,4,5,6}

twice the total number of Roast Beef sandwiches the deli sold in the 5 months

Step-by-step explanation:

Define Rb1, Rb2, Rb3, Rb4, Rb5 as

Rb1 = # of roast beef sandwiches the deli sold in month 1

Rb2 = # of roast beef sandwiches the deli sold in month 2

Rb3 = # of roast beef sandwiches the deli sold in month 3

Rb4 = # of roast beef sandwiches the deli sold in month 4

Rb5 = # of roast beef sandwiches the deli sold in month 5

Then,

# of tuna sandwiches sold in month 1 = 3 Rb1,

# of tuna sandwiches sold in month 2 = 3 Rb2,

# of tuna sandwiches sold in month 3 = 3 Rb3,

# of tuna sandwiches sold in month 4 = 3 Rb4,

# of tuna sandwiches sold in month 5 = 3 Rb5.

The deli sold

(3 Rb1 + 3 Rb2 + 3 Rb3 + 3 Rb4 + 3 Rb5) - (Rb1 + Rb2 + Rb3 + Rb4 + Rb5)  =

2 Rb1 + 2 Rb2 + 2 Rb3 + 2 Rb4 + 2 Rb5 =

2 (Rb1 + Rb2 + Rb3 + Rb4 + Rb5) =

twice the total number of Roast Beef sandwiches the deli sold in the 5 months