The alleles for eye color in a certain male fruit fly are R

The alleles for eye color in a certain male fruit fly are (R, W). The alleles for eye color in the mating female fruit fly are (R, W). Their offspring receive one allele for eye color from each parent. If an offspring ends up with either (W, W), (R, W), or (W, R), its eyes will look white. Let X equal the number of offspring having white eyes. Let Y equal the number of white-eyed offspring having (R, W) or (W, R) alleles.

(a) If the total number of offspring is n = 400, how is X distributed?

(b) Give the values of E(X) and Var(X).

(c) Given that X = 300, how is Y distributed?

(d) Give the value of E(Y | X = 300) and the value of Var(Y | X = 300).

Solution

Let X be the random variable that number of offspring having white eyes.

Let Y equal the number of white-eyed offspring having (R, W) or (W, R) alleles.

given that n = 400

There are four possible outcomes for X.

(R,W) (W,W) (W,R) (W,W)

We are given the condition that if an offspring ends up with either (W, W), (R, W), or (W, R), its eyes will look white.

Probability that number of offspring having white eyes is 1/4

X ~ Binomial (n=400, p=1/4)

E(X) = n*p = 400 * 1/4 = 100

Var(X) = n*p*(1-p) = 100*1/4*(1-1/4) = 18.75