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

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Random Variables and Distributions

Distribution of a random variableBinomial and Poisson distributionsNormal distributions

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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

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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

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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)

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Example: Birth Order of Children

** pdf: Table 7.1 on page 163

** histogram of pdf: Figure 7.1

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Important Features of a Distribution

Overall pattern Central tendency – mean Dispersion – variance or standard deviation

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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:

i

ii px

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Variance & Standard Deviation

Notations as before Variance of X:

Standard deviation (sd) of X:

i

ii pxXV 22 )()(

i

ii px 2)(

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Example: Birth Order of Children

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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.

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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 trialsFor x = 0, 1, 2, …,n,

P(X=x) = xnx pp

xnxn

)1()!(!

!

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Mean & Variance of a Binomial R.V.

Notations as before

Mean is

Variance is

np)1(2 pnp

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

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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.

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Brief Minitab Instructions

Minitab:Calc>> Probability Distributions>> Poisson;Click ‘probability’ , ‘input constant’ and x

Minitab Output:Poisson with mean = 2.4

x P( X = x )1 0.217723

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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].

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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 ∞

Read the normal distribution handout.

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Standard Normal Distribution

X: a normal r.v. with mean and standard deviation

Thenis a normal r.v. with mean 0 and standard deviation 1; called a standard normal r.v.

XZ

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 = 1P( 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

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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 = 129 mm Hg and = 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?