Binomial Theorem Mgf

Evidently the first derivative of the moment generating function evaluated at 0 is the expectation of the random variable! We really are generating moments! This result will hold in the continuous case as well. Compute the Expected Value of a Binomial Random Variable Bernoulli is special case

For any random variable X, the moment generating function MGF M Xs is M Xs E h esX i. 1 Theorem Sum of Bernoulli Binomial X 1, , X N be a sequence of i.i.d. Bernoulli random variables with parameter p Z X 1 X N Then Z is a binomial random variable with parameters N,p.

The mean and the variance of a random variable X with a binomial probability distribution can be difficult to calculate directly. Although it can be clear what needs to be done in using the definition of the expected value of X and X 2, the actual execution of these steps is a tricky juggling of algebra and summations.An alternate way to determine the mean and variance of a binomial

Note that the expected value of a random variable is given by the first moment, i.e., when 92r192.Also, the variance of a random variable is given the second central moment.. As with expected value and variance, the moments of a random variable are used to characterize the distribution of the random variable and to compare the distribution to that of other random variables.

the mgf of the binomial. If X is binomial with n trials and probability p of success, then we can write it as a sum of the outcome of each trial X Xn j1 Xj where Xj is 1 if the jth trial is a success and 0 if it is a failure. The Xj are independent and identically distributed. So the mgf of X is that of Xj raised to the n. MX j t EetX

Example 3.4 Binomial mgf The binomial mgf is MXt Xn x0 etx n x px1 pn x Xn x0 petx1 pn x The binomial formula gives Xn x0 n x uxvn x uvn Hence, letting u pet and v 1 p, we have MXt pet 1 pn The following theorem shows how a distribution can be characterized. Theorem 3.4 Let FXx and FY y be two cdfs all

Theorem. Let X be a discrete random variable with a binomial distribution with parameters n and p for some n 92in 92N and 0 92le p 92le 1 X 92sim 92Binomial n p

The Moment Generating Function of the Binomial Distribution Consider the binomial function 1 bxnp n! x!nx! pxqnx with q1p Then the moment generating function is given by 2 M xt Xn x0 ext n! Another important theorem concerns the moment generating function of a sum of independent random variables 16 If x fx

In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions.There are particularly simple results for the moment

92begingroup It makes sense to me that the Binomial Theorem would be applied to this, I'm just having a hard time working out how they get to the final result using it 92 92endgroup - CoderDake Commented Nov 13, 2012 at 2102