Find Probability density function on an arbitrary day

Question

The waiting time of a bus as T(measured in mins). Assume on weekdays (Mon to Fri), T is an exponential random variable with mean 3 mins. On weekends (Sat to Sun), T is an exponential random variable with mean 5 mins.

Find the probability density function of T on a arbitrary day.

I know how to find pmf, but if I need to find pdf, what should I do?

Answer 1

You just need to combine the two pdfs that you are given. For a random day, there is a $\frac 27$ chance it is a weekend. Use that to weight the two pdfs.

Answer 2

For any arbitrary day the random variable T will have a mixture distribution with pdf

$f(t) = \frac{5}{7}f_1(t) + \frac{2}{7}f_2(t)$

Where $f_1(t)$ and $f_2(t)$ are the pdfs of exponential distribution with mean 3 and 5 respectively