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Introduction to Probability Distributions

QSCI 381 – Lecture 12(Larson and Farber, Sect 4.1)

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Learning objectives Become comfortable with variable

definitions Create and use probability

distributions

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Random Variables-I 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

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Random Variables - II 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).

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Random Variables - III 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

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The number of scales forming the lateral line of a fish? Discrete random variable

Max swimming speed of a blue whale? Continuous random variable

Reproductive events of a Fathead minnow each year? Discrete random variable

Random Variables - III

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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.

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Probability Distributions (Discrete-I)

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.

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Probability Distributions (Discrete-II)

Mathematically: Let the possible values for the random

variable be: By the first condition:

By the second condition:

{ : 1,..., }ix i N

0 ( ) 1iP x

1

( ) 1N

ii

P x

The probability of each value of the discrete random

variable is between 0 and 1,

inclusive.

The sum of all the probabilities

is 1.

Upper summation limit

Lower summation limit

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Probability Distributions (Discrete-III)

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; 3-mature.

0 10.5

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Probability Distributions (Discrete-III)

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; 3-mature.

0 1

Does this satisfy both discrete probability conditions?

0.5

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Probability Distributions (Discrete-III)

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; 3-mature.

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.

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Probability Distributions (Discrete-IV)

Discrete probability distributions are often represented using histograms.

0.05

0.52

0.43

0

0.1

0.2

0.3

0.4

0.5

0.6

Calf Immature Mature

Maturity State

Pro

ba

bili

ty

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Example-I 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:

Verify that this is a discrete random variable.

1 2 3 4 50.02 0.05 0.1 0.2 0.36 7 8 9 10

0.23 0.05 0.02 0.02 0.01

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Means, Variances and Standard Deviations-I

The of a discrete random variable is given by:

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.

1

( )N

i ii

x P x

Each value of x is multiplied by its corresponding

probability and the products are

added.

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Means, Variances and Standard Deviations-II

The of a discrete random variable is given by:

and the by:

2 2

1

( ) ( )N

i ii

x P x

2

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Calculation of Variance (Example)

1 0.16 -1.94 3.764 0.602

2 0.22 -0.94 0.884 0.194

3 0.28 0.06 0.004 0.001

4 0.20 1.06 1.124 0.225

5 0.14 2.06 4.244 0.594

1.616

x ( )P x x 2( )x 2( )( )P x x

Hint: First find the mean, =2.94The variance is

the summation of

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Expected Value The of a discrete

random variable is equal to the mean of the random variable, i.e.:

1

( ) ( )N

X i ii

E X x P x