Question-61

Let \(X_1, X_2, \ldots, X_5\) be independent and identically distributed normal random variables with mean 0 and variance 1. Define \(Y = \sum_{i=2}^{5}X_i^2\). Which of the following statements is true about \(Y?\)

• \(Y \sim \chi^2_4\)

• \(Y \sim \chi^2_5\)

• \(Y \sim N(5,4)\)

• \(Y \sim \chi^2_6\)

Hint

Use the definition of Chi-square distribution as

If \(Z_1, Z_2, \ldots, Z_k\) be independent and identically distributed normal random variables with mean 0 and variance 1, then the random variable

\(Y = \sum_{i=1}^{k}Z_i^2\) follows a chi-square distribution with \(k\) degrees of freedom.

Answer

• \(Y \sim \chi^2_4\)

• \(Y \sim \chi^2_5\)

• \(Y \sim N(5,4)\)

• \(Y \sim \chi^2_6\)

Solution

Given, Let \(X_1, X_2, \ldots, X_5\) be independent and identically distributed normal random variables with mean 0 and variance 1. Then, \(Y = \sum_{i=2}^{5}X_i^2 = X_2^2 + X_3^2 + X_4^2 + X_5^2\) will follow chi-square distribution with 4 degrees of freedom.

Hence, option (A) is correct.