Uniform Probability Distribution 1

Question

A manager of a department store reports that the time of a customer on the second floor must wait for the elevator has a uniform distribution ranging from 2 to 4 minutes. If it takes the elevator 30 seconds to go from floor to floor, find the probability that a hurried customer can reach the first floor in less than 2.75 minutes after pushing the elevator button on the second floor.

At First I was thinking that it is just .5 integrated over [2,4] but that was incorrect

I have unlimited submission attempts so let the answers pour out!

Thanks

Answer 1

Think about it. It takes the elevator $2$ to $4$ minutes to arrive (uniformly distributed), then another $0.5$ minutes to reach the destination. You want to know the probability of reaching the destination within $2.75$ minutes of pressing the button. So you want to know the probability that the elevator arrives before what time?

Find: $\mathsf P(X \leq x)$ when $X\sim\mathcal{U}[2,4]$ and $x =\underline{\qquad}$.

Do you know the cumulative distribution function ( CDF ) of a continuous uniform distribution?

Answer 2

Hint: to arrive in less than $2.75$ minutes to the first floor, he has to wait the elevator at the second floor for a maximum of $2.25$ minutes. Which is the proportion of the area of the probability function that is identified by this value?