Question

The wait times at two banks X and Y are modeled in terms of exponential densities, each with a mean of 4 minutes.

Obtain the pdf of Z=X / Y.

Answer 1

To find the pdf of Z = X / Y, where X and Y are random variables representing the wait times at two banks with exponential densities, we can use the transformation method by finding the cumulative distribution function (CDF) of Z and then differentiating it to find the probability density function (pdf) of Z.

Step-by-step explanation:

To find the pdf of Z = X / Y, where X and Y are random variables representing the wait times at two banks, we can use the transformation method for probability density functions.

1. First, we need to find the cumulative distribution function (CDF) of Z.
2. The CDF of Z is given by P(Z <= z) = P(X / Y <= z) = P(X <= zY).
3. Since X and Y have exponential densities with mean values of 4 minutes, we can use the exponential CDF to find P(X <= t) and P(Y <= t), where t = zY.
4. By differentiating the CDF of Z with respect to z, we can find the probability density function (pdf) of Z.
5. Finally, we can substitute the value of Z into the derived pdf to obtain the pdf of Z.