The PMF for the result of any one roll of a three sided die with faces numbered

Question

The PMF for the result of any one roll of a three sided die with faces numbered 1.2 and 3 is Consider a sequence of six independent rolls of this die, and let X,- be the random variable corresponding to the result of the i^thr oll. What is the probability that exactly two of the rolls have result equal to 2? What is the probability that the first roll is 1, given that exactly three of the six rolls have result of 1? We are told that exactly three of the rolls resulted in 1 and exactly three resulted in 2. Given this information, what is the probability that the sequence of rolls is 121212? Conditioned on the event that at least one roll resulted in 3, find the conditional PMF of the number of 3s.

Explanation / Answer

PMF of Y f(y) = 2/7 if y=0

= 2/7 if y=1

= 3/7 if y=2

= 0 o.w

PMF of X f(x) = 1/7 if x=0

= 2/7 if x=2

=1/7 if x=3

=2/7 if x=4

=1/7 if x=5

E(X) = 2*2/7 + 3*1/7 + 4*2/7 +5*1/7 =16/7 = 2.28

For x=2 E(y/x) is maximum

For y=2 V(x/y) is maximum

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