1 / 24

If we can reduce our desire, then all worries that bother us will disappear.

If we can reduce our desire, then all worries that bother us will disappear. Random Variables and Distributions. Distribution of a random variable Binomial and Poisson distributions Normal distributions. What Is a Random Variable?.

milo
Download Presentation

If we can reduce our desire, then all worries that bother us will disappear.

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. If we can reduce our desire, then all worries that bother us will disappear.

  2. Random Variables and Distributions Distribution of a random variable Binomial and Poisson distributions Normal distributions

  3. What Is a Random Variable? • The numerical outcome of a random circumstance is called a random variable. Eg. Toss a dice: {1,2,3,4,5,6} Height of a student • A random variable (r.v.) assigns a number to each outcome of a random circumstance. Eg. Flip two coins: the # of heads

  4. Types of Random Variables • A continuous random variable can take any value in one or more intervals. ** eg. Height, weight, age • A discrete random variable can take one of a countable list of distinct values. ** eg. # of courses currently taking

  5. Distribution of a Discrete R.V. • X = a discrete r.v. • x = a number X can take • The probability distribution function (pdf) of X is: P(X = x)

  6. Example: Birth Order of Children ** pdf: Table 7.1 on page 163 ** histogram of pdf: Figure 7.1

  7. Important Features of a Distribution • Overall pattern • Central tendency – mean • Dispersion – variance or standard deviation

  8. Calculating Mean Value • X = a discrete r.v. • { x1, x2, …} = all possible X values • pi is the probability X = xi where i = 1, 2, … • The mean of X is:

  9. Variance & Standard Deviation • Notations as before • Variance of X: • Standard deviation (sd) of X:

  10. Example: Birth Order of Children

  11. Bernoulli and Binomial Distributions • A Bernoulli trial is a trial of a random experiment that has only two possible outcomes: Success (S) and Failure (F). The notational convention is to let p = P(S). • Consider a fixed number n of identical (same P(S)), independent Bernoulli trials and let X be the number of successes in the n trials. Then X is called a binomial radon variable and its distribution is called a Binomial distribution with parameters n and p. Read the handout for bernoulli and binomial distributions.

  12. PDF of a Binomial R.V. • p = the probability of success in a trial • n = the # of trials repeated independently • X = the # of successes in the n trials For x = 0, 1, 2, …,n, P(X=x) =

  13. Mean & Variance of a Binomial R.V. • Notations as before • Mean is • Variance is

  14. Brief Minitab Instructions • Minitab: Calc>> Probability Distributions>> Binomial; Click ‘probability’ , ‘input constant’ and n, p, x • Minitab Output: Binomial with n = 3 and p = 0.29 x P( X = x ) 2 0.179133

  15. The Poisson Distribution • a popular model for discrete events that occur rarely in time or space such as vehicle accident in a year • The binomial r.v. X with tiny p and large n is approximately a Poisson r.v.; for example, X = the number of US drivers involved in a car accident in 2008 Read the Poisson distribution handout.

  16. Brief Minitab Instructions • Minitab: Calc>> Probability Distributions>> Poisson; Click ‘probability’ , ‘input constant’ and l,x • Minitab Output: Poisson with mean = 2.4 x P( X = x ) 1 0.217723

  17. Distribution of a Continuous R.V. • The probability density function (pdf) for a continuous r.v. X is a curve such that P(a < X <b) = the area under it over the interval [a,b].

  18. Normal Distribution • Its density curve is bell-shaped • The distribution of a binomial r.v. with n=∞ • The distribution of a Poisson r.v. with l=∞ Read the normal distribution handout.

  19. Standard Normal Distribution • X: a normal r.v. with mean m and standard deviation s • Thenis a normal r.v. with mean 0 and standard deviation 1; called a standard normal r.v.

  20. Brief Minitab Instructions • Minitab: Calc>> Probability Distributions>> Normal; Click what are needed • Minitab Output: Inverse Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1 P( X <= x ) x 0.95 1.64485 Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1 x P( X <= x ) 1.64485 0.950000

  21. Example: Systolic Blood Pressure • Let X be the systolic blood pressure. For the population of 18 to 74 year old males in US, X has a normal distribution with m = 129 mm Hg and s = 19.8 mm Hg. • What is the proportion of men in the population with systolic blood pressures greater than 150 mm Hg? • What is the 95-percentile of systolic blood pressure in the population?

More Related