Question

fast - food restaurants pride themselves on being able to fill orders quickly. a study was done at a local chain to determine how long it took customers to receive their orders at the drive - through. it was discovered that the time it takes for orders to be filled is exponentially distributed with a mean of 1.5 minutes. what is the probability density function for the time it takes to fill an order?
a. (f(x)=\frac{1}{0}e^{-\frac{x}{1}})
b. none of these choices.
c. (f(x)=\frac{1}{1.5}e^{-\frac{x}{1.5}})
d. (f(x)=\frac{2}{3}e^{-\frac{2x}{3}})

Explanation:

Step1: Recall exponential - PDF formula

The probability density function of an exponential distribution is $f(x)=\lambda e^{-\lambda x}$, for $x\geq0$, and the mean of an exponential distribution is $\mu=\frac{1}{\lambda}$.

Step2: Find the value of $\lambda$

Given $\mu = 1.5$. Since $\mu=\frac{1}{\lambda}$, then $\lambda=\frac{1}{\mu}$. Substituting $\mu = 1.5=\frac{3}{2}$, we get $\lambda=\frac{2}{3}$.

Step3: Write the PDF

The probability density function is $f(x)=\frac{2}{3}e^{-\frac{2}{3}x}$ for $x\geq0$.

Answer:

d. $f(x)=\frac{2}{3}e^{-\frac{2}{3}x}$