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Could you please answer my following question related to an exponential random variable? Thank you.

Let X represent the waiting times at a telephone in an office. Assume that X is an exponential random variable with parameter λ: P(X < t) = 1 - e^{-λ*t}

At each time when the telephone rings, a staff at the ofice will toss up a coin. If the coin comes up a tail, she will pick up the phone. If the coin comes up a head, she finishes her job.

Let Y represent the outcomes of each toss. Y = 0 if the coin comes up a tail; Y = 1 if the coin comes up a head. Asume that Y is a random variable. P(Y=0) = 1-p; P(Y = 1) = p where p is a constant (0<= p <= 1).

Let T be the time that she finishes her job.

Is it correct? P(T < t) = ( 1 - e^{-λ*t} ) * p