The next four problems concern Random Variables. You must describe the random va

Question

The next four problems concern Random Variables. You must describe the random variable X by

(a) listing the values in the sample space S;

(b) listing the values in Range(X); and

(c) giving the distribution by listing the values f (x) for each x in Range(X).

For example, if the random variable were X = number of heads after flipping a single fair coin, we would have:

Problem 7. Describe the random variable X = "the number of face cards in a hand of 5 cards randomly selected without replacement from a normal deck of cards"

Hint: I showed you how to do these kinds of problems at the beginning of the lecture on Thursday. There are six patterns of outcomes, corresponding to the sets of patterns of face and non-face cards, and you'll have to count the number of hands for each pattern set; you can do this with permutations or with combinations. Be sure to check your work by summing all the probabilities; they must of course sum to 1.0.

Explanation / Answer

Here X = the number of face cards in a hand of 5 cards randomly selected without replacement from a normal deck of cards

(a) listing the values in the sample space S

= (0F,5N), (1F,4N), (2F,3N) , (3F,2N) , (4F,1N) , (5F,0N)

so here F = number of face card and N = number of non face cards

(b) Range of X E [0,5 ]

(c) Total number of face cards in standard deck = 12

Total number of non face cards = 40

f(x =0) = 5C0   * (40/52)* (39/51) * (38/50) * (37/49) * (36/48) = 0.2532

f(x=1) = 5C1 * (40/52) * (39/51) * (38/50) * (37/49) * (12/48) = 0.4220

f(x= 2) = 5C2 * (40/52) * (39/51) * (38/50) * (12/49) * (11/48) = 0.2509

f(x=3) = 5C3 * (40/52) * (39/51) * (12/50) * (11/49) * (10/48) = 0.066

f(x=4) = 5C4 * (40/52) * (12/51) * (11/50) * (10/49) * (9/48) = 0.0076

f(x=5) = 5C5 * (12/52) * (11/51) * (10/50) * (9/49) * (8/48) = 3.047 x 10-4

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