Which distribution would this be?

**Allan89a** · Dec 23rd 2009, 10:12 AM

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?

**matheagle** · Dec 23rd 2009, 09:20 PM

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.

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

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

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

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

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

**Allan89a** · Dec 24th 2009, 04:25 AM

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

**matheagle** · Dec 24th 2009, 04:51 AM

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.

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