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1
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 trials
For x = 0, 1, 2, …,n,
P(X=x) = xnx pp
xnx
n
)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.
X
Z
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
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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?