Understanding Central Moments

Central moments are statistical measures that describe the shape and characteristics of a probability distribution. The \( k \)-th central moment of a random variable \( X \) is defined as the expected value of \( (X - E[X])^k \), where \( E[X] \) is the mean of \( X \).

The question asks for the fourth central moment of a normal distribution \( X \sim N(\mu, \sigma^2) \). Here, \( \mu \) represents the mean and \( \sigma^2 \) represents the variance of the distribution. The standard deviation is \( \sigma \).

Central Moments of a Normal Distribution

The central moments for a normal distribution \( N(\mu, \sigma^2) \) follow a specific pattern:

• For odd orders \( k \), the central moment is zero: \( \mu_k = E[(X-\mu)^k] = 0 \). This is due to the perfect symmetry of the normal distribution around its mean.
• For even orders \( k \), the central moment is given by the formula: \( \mu_k = \sigma^k (k-1)!! \)

The double factorial \( (k-1)!! \) is the product of all positive odd integers up to \( k-1 \).

Calculating the Fourth Central Moment (\( \mu_4 \))

We need to find the fourth central moment, so we use the formula for an even order moment with \( k = 4 \):

\( \mu_4 = \sigma^4 (4-1)!! \)

First, let's calculate the double factorial \( (4-1)!! \):

\( (4-1)!! = 3!! \)

The double factorial of 3 (3!!) is the product of positive odd integers less than or equal to 3:

\( 3!! = 3 \times 1 = 3 \)

Now, substitute this value back into the formula for \( \mu_4 \):

\( \mu_4 = \sigma^4 \times 3 \)

\( \mu_4 = 3\sigma^4 \)

Thus, the fourth central moment of a normal distribution is \( 3\sigma^4 \).

Summary of Normal Distribution Central Moments

Revision Table: Key Moments of Normal Distribution

Additional Information: Importance of Moments

Moments, especially central moments, are crucial in statistics for characterizing the shape, spread, and other properties of a probability distribution:

• The first central moment is always 0.
• The second central moment is the variance, which measures the spread of the data around the mean.
• The third central moment is related to the skewness, which measures the asymmetry of the distribution. For a normal distribution, skewness is 0 because it is perfectly symmetric.
• The fourth central moment is related to kurtosis, which measures the "tailedness" or peakedness of the distribution. For a normal distribution, the kurtosis (\( \mu_4/\sigma^4 \)) is 3. Excess kurtosis (\( \mu_4/\sigma^4 - 3 \)) is 0 for a normal distribution.