Question

The probability distribution of the random variable X is given by
\begin{tabular}{|l|l|l|l|l|} \hline X & 0 & 1 & 2 & 3 \\ \hline P(X) & 0.2 & k & 2k & 2k \\ \hline \end{tabular}
The variance of the random variable X is

• A. \(\frac{764}{625}\)
• B. \(\frac{1}{625}\)
• C. 1
• D. \(\frac{108}{25}\)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:

This question requires calculating the variance of a discrete random variable based on its probability distribution. The process involves first determining the unknown constant 'k' by ensuring the sum of all probabilities equals 1. Subsequently, the mean (Expected Value, \( E[X] \)) and the expected value of the squared variable (\( E[X^2] \)) are computed. Finally, the variance is calculated using the standard formula.

Step 2: Key Formula or Approach:

1. Sum of probabilities: \( \sum P(X_i) = 1 \).
2. Mean (Expected Value): \( E[X] = \mu = \sum X_i P(X_i) \).
3. Expectation of \( X^2 \): \( E[X^2] = \sum X_i^2 P(X_i) \).
4. Variance: \( \text{Var}(X) = \sigma^2 = E[X^2] - (E[X])^2 \).

Step 3: Detailed Explanation:

Find k:
The sum of probabilities must be 1. \[ 0.2 + k + 2k + 2k = 1 \] \[ 0.2 + 5k = 1 \] \[ 5k = 0.8 \implies k = \frac{0.8}{5} = 0.16 \] The completed probability distribution table is:

X	0	1	2	3
P(X)	0.2	0.16	0.32	0.32

Calculate Mean \( E[X] \):
\[ E[X] = (0 \times 0.2) + (1 \times 0.16) + (2 \times 0.32) + (3 \times 0.32) \] \[ E[X] = 0 + 0.16 + 0.64 + 0.96 = 1.76 \]

Calculate \( E[X^2] \):
\[ E[X^2] = (0^2 \times 0.2) + (1^2 \times 0.16) + (2^2 \times 0.32) + (3^2 \times 0.32) \] \[ E[X^2] = (0 \times 0.2) + (1 \times 0.16) + (4 \times 0.32) + (9 \times 0.32) \] \[ E[X^2] = 0 + 0.16 + 1.28 + 2.88 = 4.32 \]

Calculate Variance \( \text{Var}(X) \):
\[ \text{Var}(X) = E[X^2] - (E[X])^2 \] \[ \text{Var}(X) = 4.32 - (1.76)^2 \] \[ \text{Var}(X) = 4.32 - 3.0976 = 1.2224 \]

Convert to Fraction:
Comparing with the provided options:
Option (A): \( \frac{764}{625} \).
Calculation: \( 764 \div 625 = 1.2224 \).
This value matches the computed variance.

Step 4: Final Answer:

The variance of the random variable \( X \) is 1.2224, which is equivalent to \( \frac{764}{625} \).