In: Statistics and Probability
We produce a random real number X through the following two-stage experiment. First roll a fair die to get an outcome Y in the set {1, 2, . . . , 6}. Then, if Y = k, choose X uniformly from the interval (0, k]. Find the cumulative distribution function F(s) and the probability density function f(s) of X for 3 < s < 4.
Answer:
Given that,
We produce a random real number X through the following two-stage experiment.
First roll a fair die to get an outcome Y in the set {1, 2, . . . , 6}.
Then, if Y = k, choose X uniformly from the interval (0, k].
The cumulative distribution function F(s) and the probability density function f(s) of X for 3 < s < 4:
We have X as U[0,k] if Y=X.
So, basically by laws of total probability,
Where, [ Probability of getting any side]
[ Since, , is pdf of U[0.k] random variable]
So,
Clearly at X=45 only terms (5,6) will remain
=0.0611