PROBABILITY DISCRETE RANDOM VARIABLES PMF (Please show steps, final solution has

Question

PROBABILITY
DISCRETE RANDOM VARIABLES
PMF
(Please show steps, final solution has been already provided, having a hard time getting to the final answer)

Monitor three phone calls and observe whether each one is a voice call or a data call. The random variable N is the number of voice calls. Assume N has PMF 0.1 n=0, 0.3 n-1,2, 3, PN (n)= (2.75) 0 othewise. Voice calls cost 25 cents each and data calls cost 40 cents each. T cents is the cost of the three telephone calls monitored in the experiment. (1) Express T as a function of V (2) Find Pr(t) and E[T]

Explanation / Answer

1) Here, as N is the number of voice calls, therefore there would be (3 - N) data calls.

Therefore the total cost for three calls would be given as:

T = 25N + 40( 3- N)

T = 120 - 15N

This is the required function of T in terms of N.

2) Now the probability density for T here is computed as:

P(T = 120 ) = 0.1 in case when n = 0,
P(T = 120 - 15 = 105 ) = 0.3, in case when n = 1,

Similarly, we get

P( T = 90 ) = 0.3
P( T = 75 ) = 0.3

This is the required PDF for T

The expected value of T here is computed as:

E(T) = 0.1*120 + 0.3*105 + 0.3*90 + 0.3*75 = 93

Therefore E(T) = 93

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