Suppose that Y is a binomial random variable with n>2 trials and success probability p. Use the

Suppose that Y is a binomial random variable with n>2 trials...

Key Concepts

Relationship Between Factorial and Ordinary Moments

The conversion between factorial moments and ordinary moments is achieved through algebraic manipulation, allowing for the derivation of one type of moment from the other. For example, expressing E(Y(Y-1)(Y-2)) in terms of E(Y^3), E(Y^2), and E(Y) demonstrates how higher-order ordinary moments can be computed by first calculating factorial moments and then using known relationships to uncover additional properties of the distribution.

Factorial Moments

Factorial moments, like E(Y(Y-1)(Y-2)), are based on the product of decreasing consecutive integers of the random variable. They are particularly useful in the analysis of discrete distributions because they often simplify the computation of moments for distributions that involve counting, such as the binomial distribution. This approach leverages combinatorial identities inherent in the structure of the distribution.

Binomial Distribution

The binomial distribution represents the probability of obtaining a fixed number of successes in a fixed number of independent trials, with each trial having two possible outcomes and a constant probability of success. It is characterized by two parameters: the number of trials (n) and the probability of success (p), and serves as a fundamental model in probability theory for discrete events.

Ordinary Moments of a Random Variable

Ordinary moments, such as E(Y), E(Y^2), and E(Y^3), provide important information about the shape and spread of a probability distribution. The third moment, for example, contributes to the understanding of skewness, while the first and second moments relate to the mean and variance respectively. They are calculated using the probability mass or density function of the random variable.

Transcript

00:01 On this problem, we want to use the technique established by theorem 3 .7, and the fact that e of y times y minus 1 times y minus 2 equals e of y cubed minus 3, e of y squared, plus 2, e of y of y of y, in order to find a formula for the expected value of y cubed.

00:30 Now first, let's work off of this left -hand side.

00:33 E of y times y minus 1 times y minus 2 is equal.

00:40 To the sum from x equals 0 to n of y times y minus 1 times y minus 2 this over down here should be y normally we use that they use way as well okay y times y minus 1 times y minus 2 times n factorial over y factorial times n minus y factorial times p to the y times q to the n minus y.

01:21 Now in order to simplify this, y, y, minus one, y minus two, we'll cancel out with part of y factorial here.

01:31 And this will give us the sum from y equals zero to n of n factorial over y minus three factorial because with the y factorial y, y, minus one, y minus two cancel it out times n minus y factorial times p to the y times to the n minus y.

02:01 Now the first three terms of this will be zero.

02:05 That's when y is zero.

02:07 That's when y is one.

02:09 And that's when y is two.

02:10 And so we can get rid of those and rewrite this then as the following.

02:18 So this is the same as the sum from y equals three to n of n factorial over y minus three factorial times n minus y.

02:35 Times p to the y times q to the n minus y.

02:45 Now again, the sum in the last expression looked very much like the binomial probabilities.

02:50 And so let's factor something out of each term here in the sum.

02:56 And so let's factor.

03:02 Let's factor out an n, an n minus 1, and an n minus 2, and a p to the third.

03:18 Now when we do this, we're left over with n minus 3 factorial over y minus 3 factorial, n minus y factorial, n minus y factorial times p to the y minus 3 times q to the n minus y.

03:41 Now from here, let's let z be y minus 3.

03:47 We're going to change our summation here...

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