Question

For $X \sim B(n, p)$, if $p = 0.6$ and $E(X) = 6$, then $\text{Var}(X) =$

• A. $6.6$
• B. $24$
• C. $2.4$
• D. $6$

Hint:

Save time by rewriting the variance formula as $\text{Var}(X) = E(X) \times (1 - p)$. This bypasses the need to compute the total number of trials ($n$) entirely!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question specifies a random variable $X$ that follows a Binomial Distribution, denoted by $X \sim B(n, p)$.
We are given the probability of success ($p$) and the expected value or mean of the distribution ($E(X)$). We need to calculate the variance ($\text{Var}(X)$).

Step 2: Key Formula or Approach:
For a Binomial Distribution, the relationships for mean and variance are uniquely defined as:
Mean: $E(X) = np$
Variance: $\text{Var}(X) = npq$
Where $q = 1 - p$ represents the probability of failure.

Step 3: Detailed Explanation:
Given the problem parameters:
$$p = 0.6$$ $$E(X) = np = 6$$ First, calculate the probability of failure ($q$):
$$q = 1 - p = 1 - 0.6 = 0.4$$ Now substitute the values directly into the variance formula. Notice that we don't even need to solve for $n$ independently because the product $np$ is already given as $6$:
$$\text{Var}(X) = (np) \times q$$ $$\text{Var}(X) = 6 \times 0.4 = 2.4$$ Alternatively, solving for $n$ gives $n = \frac{6}{0.6} = 10$. Then, $\text{Var}(X) = 10 \times 0.6 \times 0.4 = 2.4$.

Step 4: Final Answer:
The variance of the distribution is $2.4$, which corresponds to option (C).