Francis Joseph Campena, Ph.D. (DLSU) Probability and Statisitcs January 25, 2017 1 / 17

Probability and StatisticsRandom VariablesDe La Salle University

Francis Joseph Campena, Ph.D.

January 25, 2017

Outline

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Definition

Definition

A random variable is a function that associates a real number toeach element of the sample space of an experiment.

Example

• In an experiment of tossing a fair coin twice, define the randomvariable X to be the number of heads in an outcome. Thepossible values of X are {0, 1, 2}.

• In an experiment of rolling a pair of dice, define the randomvariable Y to be the sum of the numbers on the top face of eachdice (or the total number of dots on the top face of each dice).The set of all possible values of Y is {2, 3, . . . , 12}.

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Types of R.V.

If a sample space contains a countable number of sample points, thenit is called a discrete sample space. If a sample space contains aninfinite number of sample points equal to the number of points on aline segment, then it is called a continuous sample space.

• A random variable is called a discrete random variable if its set ofpossible outcomes is countable.

• A random variable is called a continuous random variable if it canassume any value in some interval or intervals of real numbers.

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Probability Distributions

Definition

The set of ordered pairs (x , f (x)) is a probability function,probability mass function or probability distribution of a discreterandom varialbe X if for each of the possible outcome x ,

1. f (x) ≥ 0.

2.∑

x f (x) = 1.

3. P(X = x) = f (x).

Consider the experiment of tossing coin twice and the random variableX defined as the number of heads in the outcome. The probabilitydistribution of X is

X = x 0 1 2

P(X = x) = f (x) 14

24

14

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Properties of PMF

The following are some important concepts in relation to theprobability distribution of a discrete random variable X .

• The values x1, x2, . . . , xk of a discrete random variable for whichis probability mass function is positive are called mass points.

• The function f (x) is usually called the probability mass function.

• To find the probability that the discrete random variable X willhave a value between a to b, we get the sum∑

a≤xi≤bf (xi ).

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Example

1 A box contains three red marbles and four green marbles. Jaypicks two marbles at random from this box. Define X to be thenumber of red marbles that jay picked. Construct a probabilitydistribution of X .

2 A shipment of 8 microcomputers to a retail outlet contains 3that are defective. If a school makes a random purchase of 2 ofthese computers, find the probability distribution for the numberof defectives that the school might have purchased.

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Definition

The function f (x) is a probability density function for thecontinuous random variable X , defined over the set of real numbers, ifthe following are satisfied:

1. f (x) ≥ 0 for all x ∈ R.

2.

∫ ∞−∞

f (x)dx = 1

3. P(a ≤ x ≤ b) =

∫ b

af (x)dx .

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Example

Let X be a continuous random variable with probability densityfunction

f (x) =

{x − 1 if 1.5 ≤ x ≤ 2.5

0 otherwise

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Properties of PDF

The following are some important concepts in relation to theprobability distribution of a continuous random variable X .

• The set of values of a continuous random variable X for whichthe value of f (x) is positive is called its support.

• The function f (x) is usually called the probability densityfunction.

• To find the probability that the continuous random variable Xwill have a value between a to b, we get the value of the integral∫ b

af (x)dx .

• The probability that a continuous random variable will take on aparticular value x is practically zero. That is, P(X = x) = 0.

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Mean and Varaince of a Discrete R.V.

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Figure:

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EXAMPLE

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EXAMPLE

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Fair Games

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Fair Games

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EXAMPLESFind the value of the constant c such that the following is a pmf of adiscrete random variable. Determine its mean and variance.

1

2

3

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