We produce a random real number X through the following two-stage experiment. First roll a fair...

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.

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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

orchestra answered 1 year ago

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