Thread: exponential random variable

1. exponential random variable

The time between patient's arriving (90 values) is

1 1 1 1 1 1 (6)
2 2 2 2 2 2 (6)
3 3 3 3 3 3 3 (7)
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 (18)
5 5 5 5 5 5 5 5 5 5 5 5 5 (13)
6 6 6 6 6 6 6 6 6 6 6 6 6 (13)
7 7 7 7 7 7 7 (7)
8 8 8 8 8 8 8 8 (8)
9 9 9 9 9 9 (6)
10 10 10 10 (4)
18 (1)

If X is an exponential random variable with mean $\theta=5$, compare the probability P(X>2) with the proportion of times that are greater than 2 from the data.

Thank you!

2. $P(X>2)={1\over 5}\int_2^{\infty}e^{-x/5}dx= e^{-2/5}$

But this is a continous rv so it also equals $P(X\ge 2)$, but your data looks discrete.

Next you're supposed to count all the times X exceeds 2 in your sample, but one could also look at all the times X exceeds 1.

This is why it's good to use the $\pm .5$ in the normal approximation to the binomial.