Scaled poisson distribution variance

Question

I have the following problem:

"Errors occur with the constant rate of 3 per hour and follow Poisson distribution.

What is the expectation and variance of errors occurred in 75 hours?'

Now expectation is easy, 3*75 = 225

I'm not sure how to go about the variance. When I try $75^2*3 -225^2$ i get a negative variance.

Answer 1

You could say

• the variance for one hour is $3$ (this is a Poisson distribution with parameter $3$)
• each hour is independent
• so the variance for $75$ hours is $75 \times 3 =225$

Or you could say the distribution over $75$ hours is also Poisson with a rate of $225$ per $75$ hours, so the variance is equal to the mean for $75$ hours of $225$