Question

Angela is going to repeatedly select cards from a standard deck of 52 cards with replacement until she gets her fifth heart card. Let Y be the number of cards she is drawing. Let X1 be the number of spot cards she draws, X2 be the number of face cards she draws, and X3 be the number of aces she draws. Find the joint pmf of Y, X1,X2, and X3.

Answer 1

In a pack of 52 cards there are 4 aces(denoted be the random variable X3), 36 spot cards(denoted be the random variable X1) and 12 face cards (denoted be the random variable X2) and Y denotes the number of cards drawn until she gets a fifth heart card. Since the draw is done with replacement all the random variables are independant. Let p denote the probability of getting a heart from the deck. Since there are 13 hearts p=13/52.

Then the marginal p.d.f of Y has negative binomial distribution and it is

The marginal pdf of X1 ,; u=0,1,2,....,

The marginal pdf of X2 ,; v=0,1,2,....,

The marginal pdf of X3 ,; w=0,1,2,...

Since x denote the total number of trials we have the relation between x , u, v, w as x=u+v+w

Hence the joint pmf of X1,X2,X3,Y are the product of their pdfs as the cards are chosen with replacement. Here number of trials (u+v+w) stops wen we get a 5th heart card

f(x,u,v,w)=f(x/u,v,w).f(u).f(v).f(w)

=;x=u+v+w; u=0,1,2,...,; v=0,1,2,....,;

w=0,1,2,....