381
Introduction to Probability
Distributions
QSCI 381 – Lecture 12
(Larson and Farber, Sect 4.1)
381

Learning objectives


Become comfortable with variable
definitions
Create and use probability distributions
Random Variables-I
381


A
X represents a
numerical value associated with each
outcome of a probability experiment.
“Random” implies that there is an element of
chance involved. Typical random variables:



Time that a tagged animal is at large.
Number of offspring produced by a female Panda
bear in one year / over her lifespan.
Length / weight of a sampled fish
Random Variables - II
381

Notation:


We will use upper case letters to denote
random variables and lower case letters to
denote particular values of random
variables.
The probability that the random variable X
has value x is therefore written as P (X=x).
Random Variables - III
381

A random variable is:



if it has a countable number of
possible outcomes that can be listed.
if it has an uncountable number
of possible outcomes (or the number of outcomes
cannot be listed). Continuous random variables
can (at least conceptually) be measured to any
degree of accuracy.
It is important to be able to distinguish
between continuous and discrete random
variables as we will analyze them differently
Random Variables - III
381

The number of scales forming the
lateral line of a fish?


Max swimming speed of a blue
whale?


Discrete random variable
Continuous random variable
Reproductive events of a Fathead
minnow each year?

Discrete random variable
381
Discrete and Continuous Random
Variables

Which of the following random variables that one
might encounter on a fishery survey are discrete and
which are continuous?







Number of animals caught in a haul.
Actual length of the 5th animal in the haul.
Length of the 5th animal in the haul as measured by an
observer.
Duration of the 7th haul.
Number of hauls before a shark is caught.
The weight of the haul.
The ratio of marketable species to unmarketable species.
Probability Distributions
(Discrete-I)
381


A
lists
each possible value the random variable can
take, together with its probability.
From our earlier discussion of probability, it
must be true that:


The probability of each value of the discrete
random variable must lie between 0 and 1.
The sum of all the probabilities must equal 1.
Probability Distributions
(Discrete-II)
381

Mathematically:


Let the possible values for the random variable
be: {xi : i  1,..., N }
The probability of
By the first condition:
each value of the
0  P( xi )  1

By the second condition:
Upper summation limit
N
 P( x )  1
Lower summation limit i 1
i
discrete random
variable is between
0 and 1, inclusive.
The sum of all the
probabilities is 1.
Probability Distributions
(Discrete-III)
381


0
Consider the random variable “what is the
maturity state of the next bowhead whale we
observe during a survey?”
This random variable is discrete because it
has three outcomes: 1-calf; 2-immature; 3mature.
0.5
1
Probability Distributions
(Discrete-III)
381


0
Consider the random variable “what is the
maturity state of the next bowhead whale we
observe during a survey?”
This random variable is discrete because it
has three outcomes: 1-calf; 2-immature; 3mature.
0.5
Does this satisfy both discrete probability conditions?
1
Probability Distributions
(Discrete-III)
381



Consider the random variable “what is the
maturity state of the next bowhead whale we
observe during a survey?”
This random variable is discrete because it
has three outcomes: 1-calf; 2-immature; 3mature.
The probabilities for each outcome are:


P(calf) = 0.05; P(immature)=0.52;
P(mature)=0.43.
The probabilities satisfy the requirements for
a discrete probability distribution.
Probability Distributions
(Discrete-IV)
381
Discrete probability distributions are
often represented using histograms.
0.6
0.52
0.5
0.43
Probability

0.4
0.3
0.2
0.1
0.05
0
Calf
Immature
Maturity State
Mature
Example-I
381


A fish is tagged and released. Given that it
is recaptured within 10 days of release, the
following are the probabilities of recapture
by day:
1
2
3
4
5
0.02 0.05
0.1
0.2
0.3
6
7
8
9
10
0.23 0.05 0.02 0.02 0.01
Verify that this is a discrete random
variable.
Means, Variances and
Standard Deviations-I
381

The
given by:
of a discrete random variable is
N
   xi P ( xi )
i 1


Each value of x is
multiplied by its
corresponding
probability and the
products are added.
Find the mean number of days before the fish in
Example I is recaptured. Note that the mean
need not be an integer.
Excel Hint: Use the “Sumproduct(A1:J1,A2:J2)” command
to find the mean.
Means, Variances and
Standard Deviations-II
381

The
of a discrete random
variable is given by:
N
   ( xi   ) 2 P( xi )
2
i 1
and the
by:
  2
Calculation of Variance
(Example)
381
x
P( x)
x
( x   )2
1
2
3
0.16
0.22
0.28
-1.94
-0.94
0.06
3.764
0.884
0.004
4
5
0.20
0.14
1.06
2.06
1.124
4.244
Hint: First find the mean,  =2.94
P( x)( x   )2
0.602
0.194
0.001
0.225
0.594
1.616
The variance is the
summation of
Expected Value
381

The
of a discrete
random variable is equal to the mean of
the random variable, i.e.:
N
E ( X )   X   xi P( xi )
i 1