Question

The variance of the Binomial distribution with probability mass function $p(x) = {}^{15}C_x \left(\frac{2}{3}\right)^x \left(\frac{1}{3}\right)^{15-x}$, $x = 0,1,\dots,15$, is _______}

• A. $\frac{5}{3}$
• B. $\frac{10}{3}$
• C. $\frac{1}{3}$
• D. $\frac{2}{3}$

Hint:

For any Binomial distribution:
Mean = $np$, Variance = $npq$.
Since $q = 1-p < 1$, the variance of a binomial distribution is always less than its mean.
Here, Mean = 10, Variance = 10/3.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the variance of a Binomial distribution characterized by the given probability mass function.

Step 2: Key Formula or Approach:
The probability mass function of a standard Binomial distribution $B(n, p)$ is:
\[ p(x) = {}^{n}C_x \cdot p^x \cdot q^{n-x} \] The formula for the variance of a Binomial distribution is:
\[ \text{Variance} = n \cdot p \cdot q \] where $n$ is the number of trials, $p$ is the probability of success, and $q = 1 - p$ is the probability of failure.

Step 3: Detailed Explanation:

• Compare the given probability mass function with the standard Binomial formula:
\[ p(x) = {}^{15}C_x \left(\frac{2}{3}\right)^x \left(\frac{1}{3}\right)^{15-x} \]
• From the comparison, identify the parameters:
- Number of trials, $n = 15$
- Probability of success, $p = \frac{2}{3}$
- Probability of failure, $q = \frac{1}{3}$

• Substitute these parameters into the variance formula:
\[ \text{Variance} = 15 \cdot \left(\frac{2}{3}\right) \cdot \left(\frac{1}{3}\right) \] \[ \text{Variance} = \frac{30}{9} = \frac{10}{3} \]

Step 4: Final Answer:
The variance of the given Binomial distribution is $\frac{10}{3}$.