1 / 22

Continuous distributions

Continuous distributions. Five important continuous distributions: uniform distribution (contiuous) Normal distribution c 2 –distribution [“ki-square”] t -distribution F -distribution. A reminder. Definition: Let X: S  R be a continuous random variable.

dyami
Download Presentation

Continuous distributions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Continuous distributions • Five important continuous distributions: • uniform distribution (contiuous) • Normal distribution • c2–distribution [“ki-square”] • t-distribution • F-distribution lecture 5

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

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

  4. Uniform distributionMean & variance • Theorem: • Let X be uniformly distributed on the interval [A,B]. • Then we have: • mean of X: • variance of X: 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 distributionwith parameters m and s2 (known). My notation: The book’s: for density function lecture 5

  6. Normal distributionExamples 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) lecture 5 m s

  7. Normal distributionMean & variance • Theorem: If X ~ N(m,s2) then • meanof X: • variance of X: Density function: N(0,1) N(1,1) N(0,2) lecture 5

  8. Normal distributionStandard 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) lecture 5

  9. Normal distributionStandard normal distribution Standard normal distribution,N(0,1),is the only normal distribution for which the distribution function is tabulated. We typically have X~N(m,s2) where m  0 and s2  1. Example:X ~ N(1,4) What is P( X> 3 ) ? lecture 5

  10. Normal distributionStandard normal distribution Theorem: Standardise If X ~ N(m,s2) then Example cont.:X ~ N(1,4). What is P(X >3) ? Z ~ N(0,1) Z ~ N(0,1) X ~ N(1,4) Equal areas = 0.1587 lecture 5

  11. Normal distributionStandard normal distribution lecture 5

  12. Normal distributionStandard normal distribution 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,m,s) Solution in R: > 1 - pnorm(3,1,2) [1] 0.1586553 Cumulative distribution function pnorm(x,m,s) lecture 5

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

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

  15. Normal distributionRelation 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(1-p)>5, then the approximation is good lecture 5

  16. Normal distributionLinear combinations Theorem: linear combinations If X1, X2,..., Xnare independent random variables, where and a1,a2,...,an are constant, then the linear combination where lecture 5

  17. The c2 distributionDefinition Definition: (alternative to Walpole, Myers, Myers & Ye) If Z1, Z2,..., Znare independent random variables, where Zi~ N(0,1), for i =1,2,…,n, then the distribution of is the c2-distribution with n degrees of freedom. Notation: Critical values: Table A.5 lecture 5

  18. The c2 distributionDefinition Definition: A continuous random variable X follows a c2-distribution with n degrees of freedom if it has density function Y ~ c2(1) Antag Y ~ c2(n) E(Y) = n Var(Y) = 2n E(Y/n) = 1 Var(Y/n) = 2/n Y ~ c2 (3) Y ~ c2 (5) lecture 5

  19. t-distributionDefinition Definition: Let Z~ N(0,1) and V~ c2(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 lecture 5

  20. t-distributionCompared to standard normal 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 lecture 5

  21. F-distribution Definition Definition: Let U~ c2(n1)andV~ c2(n2)be two independent random variables. Then the distribution of is called the F-distribution with n1 and n2 degrees of freedom. Notation:F ~ F(n1, n2) Critical values: Table A.6 lecture 5

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

More Related