## Monday, October 21, 2019

### Moment Generating Function for Binomial Distribution

Moment Generating Function for Binomial Distribution 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 X2, 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 distribution is to use the moment generating function for X. Binomial Random Variable Start with the random variable X and describe the probability distribution more specifically. Perform n independent Bernoulli trials, each of which has probability of success p and probability of failure 1 - p. Thus the probability mass function is f (x) C(n , x)px(1 Ã¢â‚¬â€œ p)n - x Here the term C(n , x) denotes the number of combinations of n elements taken x at a time, and x can take the values 0, 1, 2, 3, . . ., n. Moment Generating Function Use this probability mass function to obtain the moment generating function of X: M(t) ÃŽ £x 0n etxC(n,x))px(1 Ã¢â‚¬â€œ p)n - x. It becomes clear that you can combine the terms with exponent of x: M(t) ÃŽ £x 0n (pet)xC(n,x))(1 Ã¢â‚¬â€œ p)n - x. Furthermore, by use of the binomial formula, the above expression is simply: M(t) [(1 Ã¢â‚¬â€œ p) pet]n. Calculation of the Mean In order to find the mean and variance, youll need to know both MÃ¢â‚¬â„¢(0) and MÃ¢â‚¬â„¢Ã¢â‚¬â„¢(0). Begin by calculating your derivatives, and then evaluate each of them at t 0. You will see that the first derivative of the moment generating function is: MÃ¢â‚¬â„¢(t) n(pet)[(1 Ã¢â‚¬â€œ p) pet]n - 1. From this, you can calculate the mean of the probability distribution. M(0) n(pe0)[(1 Ã¢â‚¬â€œ p) pe0]n - 1 np. This matches the expression that we obtained directly from the definition of the mean. Calculation of the Variance The calculation of the variance is performed in a similar manner. First, differentiate the moment generating function again, and then we evaluate this derivative at t 0. Here youll see that MÃ¢â‚¬â„¢Ã¢â‚¬â„¢(t) n(n - 1)(pet)2[(1 Ã¢â‚¬â€œ p) pet]n - 2 n(pet)[(1 Ã¢â‚¬â€œ p) pet]n - 1. To calculate the variance of this random variable you need to find MÃ¢â‚¬â„¢Ã¢â‚¬â„¢(t). Here you have MÃ¢â‚¬â„¢Ã¢â‚¬â„¢(0) n(n - 1)p2 np. The variance ÃÆ'2 of your distribution is ÃÆ'2 MÃ¢â‚¬â„¢Ã¢â‚¬â„¢(0) Ã¢â‚¬â€œ [MÃ¢â‚¬â„¢(0)]2 n(n - 1)p2 np - (np)2 np(1 - p). Although this method is somewhat involved, it is not as complicated as calculating the mean and variance directly from the probability mass function.