Thread: Which distribution would this be?

Someone receives 0,1 or 2 letters to his home per day with Probability 0.2, 0.7, and 0.1 respectively.

If a year consists of 240 days in which he can receive letters, how many days are needed to get at least 99 letters with a probability of 0.99?

Which distribution would this be? How should you approximate to solve the problem?

You cna approximate this via the Central Limit Theorem or chebyshev's.
This is a multinomial, you can calculate the mean and variance easily.
The question seems to be, calculate n and I'd use the CLT.

$\displaystyle P(\sum_{i=1}^nX_i\ge 99)$

where $\displaystyle X_i$ is the number of letters she receives on day i.
and each $\displaystyle X_i$ has that distribution you stated
Next calculate the mean and variance and you can divide by n if you wish........

$\displaystyle P(\sum_{i=1}^nX_i\ge 99)=P(\bar X\ge {99\over n})$
and you may want a correction factor adjustment of 98.5 here

$\displaystyle \approx P\biggl(Z>{{98.5\over n}-\mu\over \sigma/\sqrt{n}}\biggr)$

You need to look up the normal percentile and then you solve for n, if I understood this correctly.

Hey, thanks for your post! But when is continuity correction needed like in this case?

Most people would ignore it, but you should understand that your rvs are integer based, 0,1 or 2. While the normal is continuous. Clearly there is a
difference between greater than 99 and the statement greater than or equal to 99. The probablity of exactly 99 letters is being ignored, which it shouldn't be.