Question 2 (18 points) A fair coin is tossed 4 times. Define X as the number of

Question

Question 2 (18 points) A fair coin is tossed 4 times. Define X as the number of Tails and Y as the maximum number of Heads in a row. For example, if the result of the 4 coin tosses is HTHH then X = 1 and Y = 2. 1. (7 points) Write down the joint distribution for X and Y. 2. (4 points) Compute the marginal distributions for each variable. 3. (7 points) Conditional on X1, what is the distribution of Y. That is, we know there is only one Tail among the 4 coin tosses, what is the distribution for the maximum number of Heads in a row

Explanation / Answer

1:

Heer X can take values 0, 1, 2, 3 and 4. And Y can take values 0, 1, 2, 3, and 4.

When 4 coins are tossed possible number of outcomes are 4*4 = 16. Each outcome is equally likley so probability of each outcome is 1/16 = 0.0625.

Following table shows all the possible outcomes and values of X and Y:

Following table shows the joint pdf in other way:

(2)

The marginal pdf of Y is:

The marginal pdf of X is:

3)

The conditional pdf of Y given X=1 :

Outcomes X Y P(X=x,Y=y) HHHH 0 4 0.0625 HHHT 1 3 0.0625 HHTH 1 2 0.0625 HTHH 1 2 0.0625 THHH 1 3 0.0625 HHTT 2 2 0.0625 HTTH 2 1 0.0625 TTHH 2 2 0.0625 HTHT 2 1 0.0625 THHT 2 2 0.0625 THTH 2 1 0.0625 HTTT 3 1 0.0625 THTT 3 1 0.0625 TTHT 3 1 0.0625 TTTH 3 1 0.0625 TTTT 4 0 0.0625

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