If we can reduce our desire,
then all worries that bother us will disappear.
1
Random Variables and
Distributions
Distribution
of a random variable
Binomial and Poisson distributions
Normal 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.



5
X = a discrete r.v.
x = a number X can take
The probability distribution function (pdf) of X
is:
P(X = x)
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



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Overall pattern
Central tendency – mean
Dispersion – variance or standard deviation
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:
   xi pi
i
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Variance & Standard Deviation


Notations as before
Variance of X:
V ( X )     ( xi   ) pi
2
2
i

Standard deviation (sd) of X:

2
(
x


)
pi
 i
i
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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) =
12
n!
x
n x
p (1  p)
x!(n  x)!
Mean & Variance of a Binomial R.V.

Notations as before

Mean is

13
Variance is
  np
  np(1  p)
2
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


15
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 l, 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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20
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Standard Normal Distribution

X: a normal r.v. with mean  and
standard deviation 
X 

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Then Z 
is a normal r.v. with

mean 0 and standard deviation 1;
called a standard normal r.v.
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



24
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?