Question

: Let X denote the result of a random experiment with the following cumulative distribution function (cdf): 0, x <1.5 | 1/ 6 , 1.5<=x < 2 | 1/ 2, 2 <= x <5 | 1 ,x >= 5

Calculate ?(1 ? ≤ 6) and ?(2 ≤ ? < 4.5)

b. Find the probability mass function (pmf) of ?

d. If it is known that the result of the experiment is integer, what is the probability that the result is 2? e. If it is given that the result of this experiment is an integer and a fair coin is tossed the number of times that the die shows, find the probability of obtaining exactly one head.

Thank you

Answer 1

b) The PMF for X is computed from the given CDF as:

P(X = 1.5) = P(1.5 <= X < 2) - P(X < 1.5) = (1/6)
P(X = 2) = (1/2) - (1/6) = (1/3)
P(X = 5) = 1 - (1/2) = (1/2)

Therefore the required PDF here is given as:
P(X = 1.5) = (1/6)
P(X = 2) = (1/3)
P(X = 5) = (1/2)

d) Now given that the result is an integer, probability that it is 2 is computed here as:

= P(X =2 ) / ( P(X = 2) + P(X = 5) )

Therefore 0.4 is the required probability here.

e) Now we know here that: P(X = 2 given an integer came ) = 0.4.
Therefore P(X = 5 given an integer came ) = 0.6

P(1 head | X = 2) = 2*0.5*0.5 = 0.5
P(1 head | X = 5) = 5*0.5⁵ = 0.15625

Therefore now using law of total addition of probability, the required probability is computed here as:

= P(X = 2 given an integer came ) P(1 head | X = 2) +  P(X = 5 given an integer came )P(1 head | X = 5)

= 0.4*0.5 + 0.6*0.15625

= 0.29375

Therefore 0.29375 is the required probability here.