# Finding a formula for a probability density function

The 75th percentile of a random variable X is the value X=k such that 75% of the observed values of X are less than k. For example, if the 75th percentile on an exam is 87, then 75% of the scores are less than 87.

1. Suppose a random variable X has the exponential probability density function

f(x) = {ce^-cx if x > 0, 0 otherwise},

where c is some positive constant. Find a formula in terms of c for the 75th percentile of X.

2. Let the random variable X represent the wait times, in hours, at a doctor's office and suppose that X has the probability density function

f(x) = {(4/5)e^(-4/5x) if x>0, 0 otherwise}

Use your formula from part (1) to find the 75th percentile of X.

## 1 Answer

• We are looking for a real number $t$ such that

$$P(X\leq t)=0.75$$ or equivalently $$\int_0^tc e^{-cx}dx = 0.75$$ $$\left[-e^{-cx}\right]_0^t = 0.75$$ $$-e^{-ct}+1 = \frac34$$ $$t=\frac{\ln 4}c.$$

• We apply the precedent result, putting $c:=\dfrac45$, to get $$t=\frac{\ln 4}{4/5}\approx1.73$$ that is about 1 hour and 44 minutes to see 75% of his patients.