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 tests the knowledge of finding the variance of a discrete random variable given its probability distribution. First, we must find the value of the unknown 'k' using the property that the sum of all probabilities in a distribution is 1. Then, we calculate the mean (Expected Value, \( E[X] \)) and the expectation of the square of the variable (\( E[X^2] \)). Finally, we use the variance 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 all probabilities must be 1. \[ 0.2 + k + 2k + 2k = 1 \] \[ 0.2 + 5k = 1 \] \[ 5k = 0.8 \implies k = \frac{0.8}{5} = 0.16 \] Now we can complete the probability distribution table:

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:
Now, let's check the options.
Option (A): \( \frac{764}{625} \).
Let's calculate this value: \( 764 \div 625 = 1.2224 \).
This matches our calculated variance.

Step 4: Final Answer:

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