Question

A random variable X has binomial distribution with n=10 and q=0.3, then variance of X is :

• A. 21
• B. 2.1
• C. 7
• D. 3

Hint:

In any Binomial distribution, the variance (\( npq \)) is always strictly less than the mean (\( np \)), since \( q < 1 \).
Here, \( \text{Mean} = 10 \times 0.7 = 7 \), and \( \text{Variance} = 2.1 \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A Binomial distribution is a discrete probability distribution defined by two parameters: the number of independent trials \( n \) and the probability of success \( p \) (or the probability of failure \( q \)).
Key Formula or Approach:
The relationship between the probabilities is:
\[ p + q = 1 \implies p = 1 - q \]
The variance of a binomial distribution is given by:
\[ \text{Variance} = npq \]

Step 2: Detailed Explanation:
We are given:
- Number of trials, \( n = 10 \)
- Probability of failure, \( q = 0.3 \)
First, we find the probability of success, \( p \):
\[ p = 1 - q = 1 - 0.3 = 0.7 \]
Now, we calculate the variance using the parameters \( n \), \( p \), and \( q \):
\[ \text{Variance} = npq = 10 \times 0.7 \times 0.3 \]
Performing the multiplication:
\[ 10 \times 0.7 = 7 \]
\[ 7 \times 0.3 = 2.1 \]
Thus, the variance of the random variable \( X \) is \( 2.1 \).

Step 3: Final Answer:
The variance of X is 2.1.