Calamity jane goes to the bank to make a withdrawal, and is equally likely to find 0 or 1 customers ahead of her. the service time of the customer ahead, if present, is exponentially distributed with parameter λ. what is the cdf of jane’s waiting time?

Step-by-step explanation:

The cumulative distribution function (CDF) F(x),"describes the probability that a random variableX with a given probability distribution will be found at a value less than or equal to x".

The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution". The probability density function is given by:

We are interested on the cumulative distribution function (CDF) of Jane’s waiting time. Let T the random variable that represent the Jane’s waiting time, and t possible value for the random variable T.

For the CDF for the waiting time is given by:

For this case we can use the total rule of probability and the conidtional probability. We have two options, find 0 or 1 customer ahead of Jane, and for each option we have one possibility in order to find the CDF, like this:

On this case the probability that we need to wait if we have o customers ahead is since if we don't have a customer ahead, so on this case Jane will not wait. And assuming that the probability of find 0 or 1 customers ahead of Jane is equal we have . And replacing we have:

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step-by-step explanation: