1. CHAPTER 5 CONTINUOUS PROBABILITY DISTRIBUTION Normal Distributions

2.  The most important of all continuous probability distribution  Used extensively as the basis for many statistical inference methods.  The probability density function is a bell-shaped curve that is symmetric about   The notation X ~ N (  2   denotes the random variable X has a normal distribution with mean  and variance  2.  A normal distribution with mean  and variance  2 = 1 is known as the standard normal distribution.

3. Jan 2009 3  Probability Density function of X ~ N     Graph E(X) =  and V(X) =  2

4. 4 In order to compute when X is a normal rv with parameters  and , we must determine However, it is not easy to evaluate this expression, so numerical techniques have been used to evaluate the integral when  and , for certain values of a and b and results are tabulated. This table is often used to compute probabilities for any other values of  and  under consideration. standard normal distribution : The normal distribution with parameter values  and . standard normal random variable (Z ) : A random variable having this distribution.

5. 5  Probability Density of Standard Normal RV  Cumulative Distribution Function

6. 6 Non standard Normal distribution When, probabilities involving X are computed using the standard normal table. When, probabilities involving X are computed by ‘standardizing’ or ‘linear transformation’. Then as a standard normal distribution. The key idea is that by standardizing, any probability involving X can be expressed as a probability involving a standard normal rv Z, and computed using the standard normal table.

7. 7 The function is the area under the standard normal density curve to the left of z

8. 8  Critical Value z  : for Z ~ N (0, 1)  denote the value on the z axis for which  of the area under the z curve lies to the right of z  CRITICAL VALUES OF THE STANDARD NORMAL Random Variable

9. 9 Thus Important

10. 10 Example 0: Evaluate

11. 11 a)What is the probability that a line width is greater than 0.62 micrometer? b) What is the probability that a line width is between 0.47 and 0.62 micrometer? P(0.47 < X < 0.63) = P(  0.6 < Z < 2.6) = P(Z < 2.6)  P(Z <  0.6) = 0.99534  0.27425 = 0.7211 Let X represent the line width Example 1:

12. 12 b) The line width of 90% of samples is below what value? Therefore and x = 0.5641.

13. 13 Let X represent the fill volume b) If all cans less than 12.1 or greater than 12.6 oz. are scrapped, what proportion of cans is scrapped? Proportion of cans scrapped = 0.00135 + 0.02275 = 0.0241

14. 14 a)At what value should the mean be set so that 99.9% of all cans exceed 12 ounces? Let X represent the fill volume

15. 15 Normal Approximation For Binomial Distribution It is possible to use the normal distribution to approximate binomial probabilities for cases in which n is large. If X is a binomial random variable, Is approximately a standard normal random variable. Therefore, probabilities computed from Z can be used to approximate probabilities for X.

16. 16 Normal Approximation For Binomial Distribution The normal approximation to the binomial distribution is good if For binomial distribution, E(X) = np and V(X) = np(1- p) For application purposes, if X is a binomial r.v and and m is a possible value of X

17. 17 Example 5 ( Exercise 3.10 pg 112) : X ~ Bin(n, p), n = 300 (large), p = 0.4. Approximate

18. 18 X ~ Bin(n, p), n = 300 (large), p = 0.4. Approximate