For the distributionX:-101p(x):0.30.50.2the third factorial moment is

Question

Prepp Team · Accepted Answer

Understanding Factorial Moments for Discrete Distributions

Factorial moments are useful measures in probability and statistics, particularly when dealing with discrete random variables. They are closely related to standard moments but often simplify calculations involving probability generating functions. The r-th factorial moment of a discrete random variable X, denoted by $\mu_{(r)}$, is defined as the expected value of the product $X(X-1)(X-2)...(X-r+1)$.

Mathematically, the r-th factorial moment is given by the formula:

$\mu_{(r)} = E[X(X-1)(X-2)...(X-r+1)]$

For a discrete probability distribution with probability mass function $p(x)$, this expectation is calculated as a sum over all possible values of X:

$\mu_{(r)} = \sum_x x(x-1)(x-2)...(X-r+1) p(x)$

Calculating the Third Factorial Moment

The question asks for the third factorial moment of the given discrete probability distribution. This means we need to calculate $\mu_{(3)}$. Using the formula with $r=3$, we get:

$\mu_{(3)} = E[X(X-1)(X-2)]$

For the given discrete distribution:

Possible values of X: -1, 0, 1
Corresponding probabilities p(x): 0.3, 0.5, 0.2

We need to calculate the product $X(X-1)(X-2)$ for each value of X and then sum the results multiplied by their respective probabilities.

Step-by-Step Calculation of $\mu_{(3)}$

Let's calculate the term $x(x-1)(x-2)$ for each value of $x$ in the distribution:

When $x = -1$: $(-1)(-1-1)(-1-2) = (-1)(-2)(-3) = -6$
When $x = 0$: $(0)(0-1)(0-2) = (0)(-1)(-2) = 0$
When $x = 1$: $(1)(1-1)(1-2) = (1)(0)(-1) = 0$

Now, we can organize these values along with the probabilities in a table to make the calculation clear.

$x$	$p(x)$	$x(x-1)(x-2)$	$x(x-1)(x-2)p(x)$
-1	0.3	-6	$(-6) \times 0.3 = -1.8$
0	0.5	0	$(0) \times 0.5 = 0$
1	0.2	0	$(0) \times 0.2 = 0$

The third factorial moment, $\mu_{(3)}$, is the sum of the values in the last column:

$\mu_{(3)} = (-1.8) + 0 + 0 = -1.8$

Result for the Third Factorial Moment

Based on the calculations, the third factorial moment for the given discrete probability distribution is -1.8.

Revision Table: Key Concepts

Concept	Definition/Formula	Application
Factorial Moment ($\mu_{(r)}$)	$E[X(X-1)...(X-r+1)]$	Measures shape of distribution, related to probability generating functions
Third Factorial Moment ($\mu_{(3)}$)	$E[X(X-1)(X-2)]$	Specific case for r=3
Expected Value ($E[g(X)]$)	$\sum_x g(x)p(x)$ for discrete X	General formula for calculating expectations
Discrete Probability Distribution	Set of possible values X and their probabilities p(x)	Foundation for calculating moments and other statistics

Additional Information on Moments

Moments are fundamental concepts in statistics used to describe the characteristics of a random variable's distribution.

Raw Moments ($\mu'_r$): The r-th raw moment is $E[X^r]$. The first raw moment ($\mu'_1$) is the mean of the distribution, $E[X]$.
Central Moments ($\mu_r$): The r-th central moment is $E[(X - \mu'_1)^r]$. The second central moment ($\mu_2$) is the variance, $E[(X - \mu'_1)^2]$.
Factorial Moments ($\mu_{(r)}$): As discussed, these are $E[X(X-1)...(X-r+1)]$. They are related to raw moments. For example:
- $\mu_{(1)} = E[X] = \mu'_1$ (The first factorial moment is the mean)
- $\mu_{(2)} = E[X(X-1)] = E[X^2 - X] = E[X^2] - E[X] = \mu'_2 - \mu'_1$
- $\mu_{(3)} = E[X(X-1)(X-2)] = E[X^3 - 3X^2 + 2X] = E[X^3] - 3E[X^2] + 2E[X] = \mu'_3 - 3\mu'_2 + 2\mu'_1$

Factorial moments are particularly convenient when working with distributions like the Poisson or Binomial distributions. They provide an alternative way to characterize the shape and properties of a distribution. Understanding the relationship between raw, central, and factorial moments helps in analyzing different aspects of a probability distribution.