Question

The random variable X has Binomial distribution with mean 4 and variance 2.4.
Find the parameters ‘n’ and ‘p’ and hence calculate the probability P[4 £ X £ 6].

Answer 1

Answer on Question #46545 – Math – Statistics and Probability

Problem.

The random variable $X$ has Binomial distribution with mean 4 and variance 2.4.

Find the parameters 'n' and 'p' and hence calculate the probability $P[4 \in X \in 6]$.

Solution:

If the random variable $X$ has Binomial distribution with parameters $n$ and $p$. Then mean

$\operatorname{E}(X) = np$ and variance $\operatorname{Var}(X) = np(1 - p)$. Hence $4 = np$ and $2.4 = np(1 - p)$. Therefore $1 - p = \frac{2.4}{4} = 0.6$ or $p = 0.4$ and from $4 = np$ we deduce $n = 10$. Then

$$\begin{array}{l} P(4 \leq X \leq 6) = P(4) + P(5) + P(6) = \binom{10}{4} 0.4^4 0.6^6 + \binom{10}{5} 0.4^5 0.6^5 + \binom{10}{5} 0.4^6 0.6^4 \\ \approx 0.563 \end{array}$$

Answer: $p = 0.4, n = 10, P(4 \leq X \leq 6) \approx 0.563$.

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