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Let $X_1,......,X_k$ be iid random variables from a Binomial distribution with n=4 and p=0.1.

Let $\bar{X} = \frac{1}{k}\sum_{i=1}^k X_i$. Find the exact value of P($\bar{X}$ $\leq np)$ for $k = 10$.

I have found out the value of this problem using the central limit theorem (CLT) using randomly generated values but this problem specifically states to not use CLT. At first, I thought to write it like this: $$\frac{1}{k}\sum_{i=1}^k P(X_i \leq np),$$ but I am not sure this is allowed.

Is this problem only possible if we randomly generated $X_1$ to $X_k$ random variables with $\operatorname{Bin}(4,0.1)$?

Ottavio
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learneverything
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    Maybe consider that $(k\overline X)\sim Bin(kn, p)$, and also $P(\overline X \le np) = P(k\overline X\le knp)$. For the exact value there are a few probabilities to add. – peterwhy Dec 07 '20 at 01:54

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Let $S$ be the sum of independent random variables $$S = \sum_{i = 1}^k X_i = k\overline X.$$

Then $S$ also follows binomial distribution: $S \sim Bin(kn, p) = Bin(40, 0.1)$. The required probability is

$$\begin{align*} P(\overline X \le np) &= P(k\overline X \le knp)\\ &= P(S \le 4)\\ &= \sum_{i=0}^4 \binom{40}{i}0.1^i(1-0.1)^{40-i}\\ &= \cdots \end{align*}$$

peterwhy
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