Normal Distribution

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Normal Distribution

Normal Distribution

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  • 1. 1.13 Normal distribution
  • 2. Normal distribution A continuous random variable X has a normal distribution and it is referred to as a normal random variable if its probability density is given by 1  ( x   ) / 2 2 2 f ( x ;  , )  for    x   2 e 2  w h ere -  <  <  an d  > 0 are th e p aram eters of th e n orm al d istrib u tion . The normal distribution is also known as Gaussian distribution because the Gaussian function is defined as 2 2  ( xb ) / c f ( x)  a e
  • 3. Normal distribution Normal probability density function for selected values of the parameter  and 2
  • 4. Mean of a Normal distribution   1 2 u / 2    ue du    f ( x ; ,  ) dx . 2 2    
  • 5. Variance of a Normal distribution  2   2    2   2 2 u / 2  ue du   2 2 u / 2 u / 2   e e du 2      0   2 0  0   2
  • 6. Moment Generating Function E(X )   V ar ( X )   2  t 2 2  t M X (t )  e 2
  • 7. Standard Normal Distribution A normal random variable with mean 0 and variance 1 is called standard normal random variable. If Z be a standard normal random variable, then the density function of Z is given by 1 2 z /2 f (z)  e ,   z 2
  • 8. Standardizing a random variable A random variable can be made standard by standardizing it. That is, if X is any random variable then X  E ( X ) is a standard random S .D . variable.
  • 9. The distribution function of a standard normal variable is z 1 2 t / 2 F (z)  2 e dt  P ( Z  z )  0 z a b The standard normal probabilities The standard normal probability F(z) = P(Z  z) F(b) - F(a) = P(a  Z  b)
  • 10. If a random variable X has a normal distribution with the mean  and the standard deviation , then, the probability that the random variable X will take on a value less than or equal to a, is given by  X  a  a a P( X  a)  P    P Z    F            Similarly b  a  P (a  X  b)  F   F       
  • 11. The Normal Distribution Although the normal distribution applies to continuous random variable, it is often used to approximate distributions of discrete random variables. For that we must use the continuity correction according to which each integer k be represented by the interval from k – ½ to k + ½. For instance, 3 is represented by the interval from 2.5 to 3.5, “at least 7” is represented by the interval from 6.5 to  and “at most 5” is represented by the interval from - to 5.5. Similarly “less than 5” is represented by the interval from - to 4.5 and “greater than 7” by the interval from 7.5 to .
  • 12. The normal Approximation to the Binomial Distribution • The normal distribution can be used to approximate the binomial distribution when n is large and p the probability of a success, is close to 0.50 and hence not small enough to use the Poisson approximation. Normal approximation to binomial distribution Theorem If X is a random variable having the binomial distribution with the parameters n and p, and if X  np Z  np (1  p ) then the limiting form of the distribution function of this standardized random variable as n   is given by
  • 13. The normal Approximation to the Binomial Distribution z 1 2 t / 2 F (z)    2 e dt    z  . Although X takes on only the values 0, 1, 2, … , n, in the limit as n   the distribution of the corresponding standardized random variable is continuous and the corresponding probability density is the standard normal density.
  • 14. The normal Approximation to the Binomial Distribution A good rule of thumb for the normal approximation For most practical purposes the approximation is acceptable for values of n & p such that Either p ≤ 0.5 & np > 5 Or p > 0.5 & n (1-p) > 5.