Stochastic Models Class There are 5 different balls with numbers from 1 to 5. Ea

Question

Stochastic Models Class

There are 5 different balls with numbers from 1 to 5. Each ball has a unique number. Consider an experiment that randomly draws one ball and then without replacement draws another, randomly. Let X be the random variable of the outcome of the first ball, and Y be the random variable of the second outcome. Answer the following: What are the following probabilities: Pr(X = i) for i = 1 ~ 5. What are the following probabilities: Pr(Y = i) for i = 1 ~ 5. What are the possible outcomes of X + Y, i.e. the range of the random variable? What are the following probabilities: Pr(X + Y = i) for all i's in the range above.

Explanation / Answer

P(X=i) = 1/5, as all the outcomes are equally likely in the first draw.

P(Y=i) = 1/4, as Y denotes the second draw and we have 4 balls now, each being equally likely.

(X+Y) can have the following values:

Value   Possible first and second draw Count

3          (1,2), (2,1)                                 2

4          (1,3), (3,1)                                 2

5          (1,4),(2,3),(3,2),(4,1)                 4

6          (1,5),(2,4),(4,2),(5,1)                 4

7          (2,5),(3,4),(4,3),(5,2)                 4

8          (3,5),(5,3)                                  2

9          (4,5),(5,4)                                  2

Total no. of outcomes = (2+2+4+4+4+2+2) = 20.

P(X+Y=3) = 2/20.

P(X+Y=4) = 2/20.

P(X+Y=5) = 4/20.

P(X+Y=6) = 4/20.

P(X+Y=7) = 4/20.

P(X+Y=8) = 2/20.

P(X+Y=9) = 2/20.

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