Question

4. Suppose you continually collect coupons and that there are two different types of coupon, type A and type B. Suppose also that each time a new coupon is obtained it is a type A coupon with probability 1/3 and a type B coupon with probability 2/3, independently of what coupons you have collected so far. Let X be the number of coupons collected until you have at least one coupon of both types. (a) Find the probability mass function of X. (b) Find E(X).

Answer 1

The probability mass function of X will be a mixture of two geometric probability mass functions. If the first coupon is Type A

1-2 1 2 1 P(X = z) = 5x5xG).

If the first coupon is Type B

2 1-2 1 x=x 2 P(X = x) =

So,

2 P(X =r). 22- 1 37 +

For expectation we first draw the first coupon and after that we calculate the expectation based on that.

E(x) = 1+ تا انت x E(Xfirst draw is B) += انت E(X|first draw is A)

E(X|first draw is A) => [Expectation of geometric is inverse of probability

E(X first draw is B) = 3 Expectation of geometric is inverse of probability

E(X) = 1+ تا | نت نت سنن اب + - = 1+2+ 0.3 = 3.5