1. lecture 5 1 Continuous distributions Five important continuous distributions: 1.uniform distribution (contiuous) 2.Normal distribution  2 –distribution[“ki-square”] 4.t-distribution 5.F-distribution

2. lecture 5 2 A reminder Definition: Let X: S  R be a continuous random variable. A density function for X, f (x), is defined by: 1. f (x)  0 for all x 2. 3. f (x) xb a P(a<X<b)

3. lecture 5 3 Uniform distribution Definition Definition: Let X be a random variable. If the density function is given by then the distribution of X is the (continuous) uniform distribution on the interval [A,B]. x A B f (x)

4. lecture 5 4 Uniform distribution Mean & variance Theorem: Let X be uniformly distributed on the interval [A,B]. Then we have: mean of X : variance of X :

5. lecture 5 5 Normal distribution Definition Definition: Let X be a continuous random variable. If the density function is given by then the distribution of X is called the normal distribution with parameters  and  2 (known). My notation: The book’s: for density function

6. lecture 5 6 Normal distribution Examples The normal distribution is without doubt the most important continuous distribution, since many phenomena are well described by it. IQ among AAU students Height among AAU students Plotting in Matlab: >> x=90:1:150; y=normpdf(x,120,10); plot(x,y)  

7. lecture 5 7 Normal distribution Mean & variance Theorem: If X ~ N( ,   ) then mean of X : variance of X : Density function: N(1,1 ) N(0,2 ) N(0,1 )

8. lecture 5 8 Normal distribution Standard normal distribution Standard normal distribution: Z ~ N(0,1) Density function: Distribution function: (see Table A.3) Notice!! Due to symmetry P(Z   z) = 1  P(Z  z)

9. lecture 5 9 We typically have X~N(   ) where  and   . Normal distribution Standard normal distribution Standard normal distribution, N(0,1), is the only normal distribution for which the distribution function is tabulated. Example: X ~ N(1,4) What is P( X  3 ) ?

10. lecture 5 10 Normal distribution Standard normal distribution Example cont.: X ~ N(1,4). What is P(X  3) ? Theorem: Standardise If X ~ N( ,  2 ) then Z ~ N(0,1) X ~ N(1,4) Equal areas = 0.1587 Z ~ N(0,1)

11. Normal distribution Standard normal distribution lecture 5 11

12. Normal distribution Standard normal distribution lecture 5 12 Example cont.: X ~ N(1,4). What is P(X  3) ? Solution in Matlab: >> 1 – normcdf(3,1,2) ans = 0.1587 Cumulative distribution function normcdf(x,,) Solution in R: > 1 - pnorm(3,1,2) [1] 0.1586553 Cumulative distribution function pnorm(x,,)

13. lecture 5 13 Normal distribution Example Problem: The lifetime of a light bulb is normal distributed with mean 800 hours and standard deviation 40 hours: 1.Find the probability that the lifetime of a bulb is between 750-850 hours. 2.Find the number of hours b, such that the probability of a bulb having a lifetime longer than b is 90%. 3.Find a time period symmetric around the mean so that the probability of a lifetime in this interval has probability 95%

14. Normal distribution Example lecture 5 14 Solution in Matlab: 1.P(750 > normcdf(850,800,40)-normcdf(750,800,40) 2.? 3.... Solution in R: 1.P(750 pnorm(850,800,40)-pnorm(750,800,40) 2.P(X > b) = 0.90  P(X  b) = 0.10 > qnorm(0.1,800,40) 3....

15. lecture 5 15 Normal distribution Relation to the binomial distribution If X is binomial distributed with parameters n and p, then is approximately normal distributed. Rule of thumb: If np  5 and n(  p)  5, then the approximation is good

16. lecture 5 16 Normal distribution Linear combinations Theorem: linear combinations If X 1, X 2,..., X n are independent random variables, where and a 1,a 2,...,a n are constant, then the linear combination where

17. lecture 5 17 The   distribution Definition Definition: (alternative to Walpole, Myers, Myers & Ye) If Z 1, Z 2,..., Z n are independent random variables, where Z i ~ N(0,1), for i =1,2,…,n, then the distribution of is the   -distribution with n degrees of freedom. Notation: Critical values: Table A.5

18. The   distribution Definition lecture 5 18 Definition: A continuous random variable X follows a   - distribution with n degrees of freedom if it has density function Y ~  2 (1) Y ~  2 (3) Y ~  2 (5) Antag Y ~  2 (n) E(Y) = n Var(Y)= 2n E(Y/n)= 1 Var(Y/n)= 2/n

19. lecture 5 19 t-distribution Definition Definition: Let Z ~ N(0,1) and V ~   (n) be two independent random variables. Then the distribution of is called the t-distribution with n degrees of freedom. Notation: T ~ t(n) Critical values: Table A.4

20. t-distribution Compared to standard normal lecture 5 20 X ~ N(0,1) T ~ T(3) T ~ T(1) The t-distribution is symmetric around 0 The t-distribution is more flat than the standard normal The more degrees of freedom the more the t- distribution looks like a standard normal

21. lecture 5 21 F-distribution Definition Definition: Let U ~  2 (n 1 ) and V ~  2 (n 2 ) be two independent random variables. Then the distribution of is called the F-distribution with n 1 and n 2 degrees of freedom. Notation: F ~ F(n 1, n 2 ) Critical values: Table A.6

22. F-distribution Example lecture 5 22 F ~ F(10,30) F ~ F(6,10) F ~ F(20,50)