Question

Choose the correct statements about chi-square probability density function.
LIST:
A. They start at 0.
B. They are Symmetrical.
C. They have a mean equal to their degree of freedom.
D. They have variance equal to degree of freedom.
E. They have variance equal to 2 $\times$ degree of freedom.

• A. A, C and E only
• B. A, B and D only
• C. A, B and C only
• D. A, C and B only

Hint:

Chi-square distribution: Mean = df, Variance = 2 $\times$ df, starts at 0, positively skewed.

Answer 1

The Correct Option is A

Step 1: General properties of chi-square distribution.
The chi-square distribution is widely used in hypothesis testing, especially in tests of independence and goodness of fit.
It is defined only for non-negative values.
Step 2: Important characteristics.
- The chi-square distribution starts at 0 (it cannot take negative values).
- It is positively skewed and not symmetrical, though it approaches normal distribution as the degree of freedom increases.
- The mean of a chi-square distribution is equal to its degrees of freedom ($df$).
- The variance of chi-square distribution is equal to $2 \times df$.
Step 3: Check each statement.
- (A) Correct, chi-square starts at 0.
- (B) Incorrect, it is not symmetrical, it is skewed.
- (C) Correct, mean = df.
- (D) Incorrect, variance is not equal to df.
- (E) Correct, variance = 2 $\times$ df.
Step 4: Conclusion.
Thus, the correct combination is A, C, and E.