Question

Suppose that Phil and Jen each have a bag containing 10 distinguishable objects. The two bags have the same content. Phil and Jen randomly choose 3 objects each from their bags. Consider the following random variables: X − the number of objects chosen by both Phil and Jen Y − the number of objects not chosen by either Phil or Jen Z − the number of objects chosen by exactly one of Phil and Jen Compute E(X), E(Y ) and E(XZ).

Answer 1

Answer:-

Given That:-

Phil and Jen each have a bag containing 10 distinguishable objects. The two bags have the same content. Phil and Jen randomly choose 3 objects each from their bags.

X - can take values 0,1, 2, -------------- 20

Y - can take values 0,1, 2, -------------- 10

Z - can take values 0,1, 2, -------------- 10

P(X = 1) = 1/20

P(X = 2) = 2/20

P(X = 10) = 1/20

= 1/2

Now

X - bags discrete distribution denoting number of objects choose by boil Phil and Jen.

Therefore

P(Xi = x) = x/20 ix = 1, 2, ----- 20

= f(x)

Similarly P(Y = y) = y/10 i, y = 1, 2, ------ 10

= f(y)

Y bags a discrete distribution denoting number of objects not chosen by either Phil or Jen, on the other hand.

Z bags a discrete distribution denoting number of object chosen by exactly one Phil and Jen

P(Z = z) = z/10, z = 1, 2, --------- 10

= f(z)

Therefore,

= 143.5

Therefore E(X) = 143.5

= 38.5

Therefore E(Y) = 38.5

= 1104.125

Therefore E(Z) = 1104.125

Now,

Assuming X and Z are independent

f(x, z) = f(x) f(z)

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