Question

If X~B(n,p) with n = 10, p = 0.4 then E(x²) =

• A. 4
• B. 2.4
• C. 3.6
• D. 18.4

Answer 1

Answer: (D)

18.4

Answer 2

To find $E(X^2)$ for a binomial distribution $X \sim B(n,p)$, we use the properties of mean and variance.

Given: $X \sim B(n,p)$ $n = 10$ $p = 0.4$

1. Calculate the mean $E(X)$: For a binomial distribution, the mean is given by $E(X) = np$. $E(X) = 10 \times 0.4 = 4$

2. Calculate the probability of failure $q$: $q = 1 - p = 1 - 0.4 = 0.6$

3. Calculate the variance $Var(X)$: For a binomial distribution, the variance is given by $Var(X) = npq$. $Var(X) = 10 \times 0.4 \times 0.6 = 4 \times 0.6 = 2.4$

4. Calculate $E(X^2)$ using the relationship between variance, mean, and $E(X^2)$: The variance is also defined as $Var(X) = E(X^2) - [E(X)]^2$. Rearranging this formula to find $E(X^2)$: $E(X^2) = Var(X) + [E(X)]^2$ Substitute the calculated values of $Var(X)$ and $E(X)$: $E(X^2) = 2.4 + (4)^2$ $E(X^2) = 2.4 + 16$ $E(X^2) = 18.4$