Probability and Statistics II week 11

From Equation (3.2.3), we see that the normal distribution has the cumulative distribution function , ……………………………………………(3.2.4) From this we obtain , ………………………………(3.2.5) The integral in (3.2.4) cannot be evaluated by elementary methods, but can be represented in terms of the integral ……………………………………………………………………(3.2.6) which is the distribution function of the normal distribution with mean = 0 and variance = 1 and has been tabulated (see mathematical tables). Infact, if we set , then , and we have to integrate from . From Eq. (3.2.4) we thus obtain drops out, and the expression on the right equal (3.2.6), where . That is ………………………………………………………………………..(3.2.7) The cumulative distribution of z is known as the standard normal distribution. From this important formula and Eq. (3.2.5) we obtain another important formula  Let X be binomial with parameters n and p. For large n, X is approximately normal with mean, and variance . For practical purposes, use normal approximation to binomial distribution for either p > 0.5 and np > 5 or p > 0.5 and n (1-p) > 5.  Let X be Poisson with parameter . Then for large values of n, X is approximately normal with mean and variance .