The third central moment of a random variable is a measure of the asymmetry or skewness of its probability distribution. It is defined as the expected value of the cube of the deviation of the random variable from its mean, mathematically represented as E[(X - μ)^3], where X is the random variable and μ is its mean.
If X and Y are two independent random variables, the third central moment of their sum (Z = X + Y) can be calculated using the formula: E[(Z - μ_Z)^3] = E[(X - μ_X + Y - μ_Y)^3]. Given the independence of X and Y, this expands and simplifies to E[(X - μ_X)^3] + E[(Y - μ_Y)^3], which means the third central moment of the sum is the sum of the third central moments of the individual variables.
The third central moment is important because it provides information about the skewness of a distribution. Skewness indicates whether the data is symmetrically distributed or if it tends to have a longer tail on one side. Positive skewness indicates a longer right tail, while negative skewness indicates a longer left tail.
For independent random variables X and Y, the third central moment has the additive property: the third central moment of their sum is the sum of their individual third central moments. This property holds because the cross-product terms involving mixed moments vanish due to the independence of the variables.
Yes, the third central moment can be zero. When it is zero, it indicates that the distribution is symmetric around its mean. This symmetry means that the tails on either side of the mean are balanced, leading to no skewness in the distribution.