Discrete Random Variables
Chapter 4 Objectives
1
The student will be able to
 Recognize and understand
discrete probability distribution
functions, in general.
 Recognize the Binomial
probability distribution and apply
it appropriately.
 Calculate and interpret expected
value (average).
2
 Types





General
Binomial
Poisson (not doing)
Geometric (not doing)
Hypergeometric (not doing)
 Calculator
becomes major tool
3
 Probability
Distribution Function
(PDF)

Characteristics




each probability is between 0 and 1,
inclusive
the sum of the probabilities is 1
An edit of the Relative Frequency
Table where the Rel Freq column is
relabeled P(X) and we drop the
Freq and Cum Freq columns
Calculated from the PDF


Mean (expected value)
Standard Deviation
An example
4
 Characteristics





each probability is between 0 and 1,
inclusive
the sum of the probabilities is 1
fixed number of trials
only 2 possible outcomes
for each trial, probabilities, p and
q, remain the same (p + q = 1)
 Other






facts
X ~ B(n, p)
X = number of successes
n = number of independent trials
x = 0,1,2,…,n
µ = np
Problem 8
σ = sqrt(npq)
5

What the calculator can do
Find P(X = x)
 Binompdf(n, p, x)
 Find P(X < x)
 Binomcdf(n, p, x)


What the calculator needs help
with
Find P(X < x) = P(X < x-1)
 Binomcdf(n, p, x-1)
 Find P(X > x) = 1 – P(X < x)
 1 – Binomcdf(n, p, x)
 Find (X > x) = 1 – P(X < x-1)
 1 – Binomcdf(n, p, x-1)

6
Continuous Random Variables
Chapter 5 Objectives
7
The student will be able to
 Recognize
and understand
continuous probability density
functions in general.
 Recognize the uniform
probability distribution and apply
it appropriately.
 Recognize the exponential
probability distribution and apply
it appropriately.
8
 Types



Uniform
Exponential
Normal
 Characteristics




Outcomes cannot be counted,
rather, they are measured
Probability is equal to an area
under the curve for the graph.
Probability of exactly x is zero
since there is no area under the
curve
PDF is a curve and can be drawn so
we use f(x) to describe the curve,
I.E. there is an equation for the
curve
9
X = a real number between a and b
 X ~ U(a, b)
 µ = (a+b)/2
 σ = sqrt((b-a)2/12)
 Probability density function
 f(x) = 1/(b – a)
 To calculate probability find the
area of the rectangle under the
curve





P (X < x) = (x - a)*f(x)
P (X > x) = (b – x)*f(x)
P (c < X < d) = (d – c)*f(x)
(we are not doing conditional
probability)
10
 Example
- The amount of time a
car must wait to get on the
freeway at commute time is
uniformly distributed in the
interval from 1 to 6 minutes.
X = the amount of time (in minutes)
a car waits to get on the freeway
at commute time
1<x<6
X ~ U(1, 6)
µ = (6 + 1)/2 = 3.50
σ = sqrt((6 – 1)2/12) = 1.44
11
 What
is the probability a car
must wait less than 3 minutes?
Draw the picture to solve the
problem.

P(X < 3) = ____________

P(2.5 < x < 5.6)

Find the 40th percentile.

The middle 60% is between what
values?
12
X ~ Exp(m)
 X = a real number, 0 or larger.
 m = rate of decay or decline
 Mean and standard deviation




PDF


f (x) = me^(-mx)
Probability calculations




µ = σ = 1/m
therefore m = 1/µ
P (X < x) = 1 – e^(-mx)
P (X > x) = e^(-mx)
P (c < X < d) = e^(-mc) – e^(-md)
Percentiles

k = (LN(1-AreaToThe Left))/(-m)
13
 An
example - Count change.
 Calculate mean, standard
deviation and graph




X = amount of change one person
carries
0<x<?
X ~ Exp( m )
µ = σ = 1/ m
 Find
P(X < $2.50), P(X > $1.50),
P($1.50 < X < $2.50),
P(X < k) = 0.90
14
The Normal Distribution
Chapter 6 Objectives
15
The student will be able to
 Recognize the normal probability
distribution and apply it
appropriately.
 Recognize the standard normal
probability distribution and apply
it appropriately.
 Compare normal probabilities by
converting to the standard
normal distribution.
16
 The

Bell-shaped curve
IQ scores, real estate prices,
heights, grades
 Notation


X ~ N(µ, σ )
P(X < x), P(X > x), P(x1 < X < x2)
 Standard

Normal Distribution
z-score


Converts any normal distribution to a
distribution with mean 0 and standard
deviation 1
Allows us to compare two or more
different normal distributions
 z = (x - µ)/ σ
Comparing
17
 Calculator

Normalcdf(lowerbound,upperbound,µ,
σ)
 if P(X < x) then lowerbound is -1E99
 if P(X > x) then upperbound is 1E99

percentiles

invNorm(percentile,µ,σ)
example
18
The Central Limit Theorem
Chapter 7 Objectives
19
The student will be able to
 Recognize
the Central Limit
Theorem problems.
 Classify continuous word
problems by their distributions.
 Apply and interpret the Central
Limit Theorem for Averages
20
 Averages


If we collect samples of size n and
n is “large enough”, calculate each
sample’s mean and create a
histogram of those means, the
histogram will tend to have an
approximate normal bell shape.
If we use X = mean of original
random variable X, and X =
standard deviation of original
variable X then
x 

X ~ N  x,

n

21
 Demonstration
of concept
 Calculator
still use normalcdf and invnorm but
need to use the correct standard
deviation.
 Normalcdf(lower,
upper,X,X/sqrt(n))

 Using
the concept
22
 What’s




Chapter 4
Chapter 5
Chapter 6
Chapter 7
 21

multiple choice questions
The last 3 quarters’ exams
 What

fair game
to bring with you
Scantron (#2052), pencil, eraser,
calculator, 1 sheet of notes (8.5x11
inches, both sides)
23