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A biostatistician is interested in studying the time X (in seconds) it takes a hematology cell...

Question

A biostatistician is interested in studying the time X (in seconds) it takes a hematology cell...

A biostatistician is interested in studying the time X (in seconds) it takes a hematology cell counter to complete a test on a blood sample. The probability density function of the aforementioned time is:

f(x) =
cos x if 0 < x < pi/2
0 elsewhere.

1. Find the cumulative distribution function and compute the median and 65-th percentile of the time to complete a test on a sample.
2. What is the percentage of tests require less than /3 seconds to complete?
3. Suppose that this biostatistician is interested in the following composite function of time to complete a test on a sample: exp(sin x). Compute the mean and variance of the underlying random variable.