[Math] Independent and identically distributed normal random variables.

Question

Successive monthly sales are independent normal random variables with mean 100 and variance 100.

Find the probability that at least one of the next 4 months has sales above 105.

I was thinking since the variables are i.i.d., then perhaps it would make sense to find the joint pdf of each one. So I call $X_i$ the number of sales in month $i$, and the first five months are $X_1,\ldots,X_5$. The joint pdf is $f(x_1,\ldots,x_5)=(\frac{1}{10\sqrt{2\pi}})^5e^{-1/200\sum_{i=1}^5(x_i-100)^2}$. But then, when I thought about it logically, finding the probability that at least one of the $X_i$ is above $105$ wouldn't really have anything to do with the joint pdf.

Could someone explain what the joint pdf would be useful for, and then also explain where I might start with actually figuring out my problem?

Answer 1

You should follow this:

1. The probability at least one is above 105 is equivalent of $ 1 - P \left( {X}_{1}, {X}_{2}, {X}_{3}, {X}_{4} \leq 105 \right) $.
2. Since all are independent $ P \left( {X}_{1}, {X}_{2}, {X}_{3}, {X}_{4} \leq 105 \right) = P \left( {X}_{1} \leq 105 \right) P \left( {X}_{2} \leq 105 \right) P \left( {X}_{3} \leq 105 \right) P \left( {X}_{4} \leq 105 \right) $.

Now, the probability of each one is known by their properties (Gaussian where the mean and variance are given).