Question

If $X$ follows binomial distribution with parameters 10 and 1/2, then $E(X - 5)^2 =$

• A. 0.25
• B. 2.50
• C. 5
• D. 10

Hint:

Recognize that $E(X - c)^2$ represents the variance whenever $c$ is equal to the mean $E(X)$. This saves unnecessary expansion of the expectation operator.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The expected squared deviation of a random variable from its mean is defined as the variance of that random variable.
Key Formula or Approach:
If $X$ is a Binomial random variable with parameters $n$ and $p$:
\[ \text{Mean}, \mu = E(X) = n \cdot p \]
\[ \text{Variance}, \sigma^2 = E(X - \mu)^2 = n \cdot p \cdot (1 - p) \]

Step 2: Detailed Explanation:
We are given:
Number of trials, $n = 10$
Probability of success, $p = 1/2$
First, let us calculate the mean of $X$:
\[ \mu = E(X) = 10 \cdot \frac{1}{2} = 5 \]
The expression we need to calculate is $E(X - 5)^2$.
Since $\mu = 5$, this expression is exactly equal to the variance of $X$:
\[ E(X - 5)^2 = E(X - \mu)^2 = \text{Var}(X) \]
Now, calculate the variance:
\[ \text{Var}(X) = n \cdot p \cdot (1-p) = 10 \cdot \frac{1}{2} \cdot \left(1 - \frac{1}{2}\right) \]
\[ \text{Var}(X) = 10 \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{10}{4} = 2.5 \]
Therefore, the expected value is 2.50.

Step 3: Final Answer
The correct option is (B).