0
$\begingroup$

Problem: A standard deck of 52 cards has two red suits (diamonds and hearts) and two black suits (clubs and spades). Each suite has 13 cards with denominations 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace. The Jack, Queen and King are known as face cards.

Three cards are drawn randomly (without replacement) from a standard deck.

a. Let the random variable X be the number of face cards in the three cards drawn. Show your calculations for finding its expected value E[X], briefly showing the steps.

b. Suppose each Jack is worth 1 point, and each Queen or King 2 points, while all other cards are worth zero points. Let the random variable Y be the sum of the points in the three cards drawn. Show your calculations for finding the expected value E[Y], quickly showing the steps.

c. Show your calculation of Var(X) and Var(Y) for the random variables described in part (a) and part (b).

d. Find the probability distribution function of Y.

I'm so far trying to understand this question and this is what I know currently...

a) The equation used is discrete random variable: $$E[X]= \displaystyle\sum_{x} x \cdot P(x)$$

I know that for each of the face cards (Jack, Queen and King) there is a $$\frac{4}{52}$$ probability at one suit. How do I show the complete calculation based on what I know?

b) This question is slightly different from part (a) and changes everything by giving a value to the face cards. So how does this change the probability and the calculation for the discrete random variable for Y?

c) Once I know this I should try to find the variance of X and Y with the following equation: $$Var(X)=E[(x-E[X])^2]$$ $$Var(Y)=E[(y-E[Y]^2]$$

d) Now the problem asks me to know for the probability distribution of Y? I assume that I should try to see if the values of Y are close to E[Y] or scattered far from E[Y].

This question seems a bit confusing and I appreciate your guidance in trying to understand the use of discrete random variable and probability distribution. I'm stuck and kind of know what sort of equations I need to use but don't know how to start. Your hindsight and opinion is welcomed. Feel free to comment in anything that I might be wrong.

1 Answers 1

0

Well actually the probabilities change each time you draw a card! So in fact, your denominator will decrease by one for each card you draw. For part a, I suggest doing a table for finding the expected value. Finding the variance should be the next part but to calculate $E[X^2$] and $E[Y^2$], just square the values you obtain from the table but not the probabilities since those are "fixed". I approached a similar problem last semester on my probability class but except it was about rolling a die and not cards.