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Stat 305, Fall 2014 Name

Chapter 5.1: Discrete Random Variables (part I)

The Basics of Random Variables

Random VariableDefinition: Arandom variableis a quantity that (prior to observation) can be thought ofas dependent on chance phenomenon. (i.e. you cant know the outcome until it happens.)

Usually use capital letters at the end of the alphabet to represent rvs; X, Y, Z

Examples:

Important features of a Random Variable (RV):

The set of possible values.

Some specification of how likely these values are to occur; i.e. probabilities foreach value.

Together these values and probabilities define the distribution of a RV.

Done differently for discrete and continuous RVs.

Discrete Random Variable

Definition: Adiscrete RV has isolated or separated possible values (rather than a contin-uum of available outcomes); i.e.: a discrete set of possible values

Examples:

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Continuous Random Variable

Definition: A continuous RV has an entire (continuous) interval of numbers as its set ofpossible values

Examples:

Probability Mass Function (PMF)

Definition: Aprobability mass function (pmf )for a discrete rv X, having possible valuesx1, x2, . . . is a nonnegative function, denoted as f(x), withf(xj) giving the probability that

X takes on the value xj .

i.e.: f(xj) =P[X=xj ]

Recall X is the random variable, x is a specific numeric value

Properties:

f(x) is in the interval [0, 1] for all x.

The values off(x) sum to 1 when the sum is taken across all possible values for

x; i.e.:

i

f(xi) = 1

Familiar examples of PMFs

1. Roll a die (6 sided)

2. Flip a fair coin

3. Flip an unfair coin

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Example 5.1

Let X = the number of goals scored by a hockey team in each of their first 9 games. Supposea team has X= 1, 1, 0, 5, 0, 2, 8, 4, 1 goals

Find the function f(x)

Find P[X >2] and P[1 X 5]

Example 5.2

For a manufacturing process that molds 6 units at a time, let X be the number of defectiveunits that result and find the following probabilities.

x 0 1 2 3 4 5 6

f(x) 0.30 0.20 0.20 0.10 0.10 0.05 0.05

1. P(X 3)

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Cumulative Density Function (CFD)

Definition: Acumulative density function, is a function F(x), that gives the probabilitythat X takes that value or a smaller one.

F(x) =P[X x] =

yx

f(y)

Another way of specifying a probability distribution. Properties:

F(x) is 0 for all x-values below x(1) and 1 for all x-values above x(n).

F(x) is non-decreasing (it can only increase or stay the same).

Example 5.1 (Hockey, continued)

Find F(x) and F(2.9)

Example 5.2 (continued)

Plot the CDF for the data.x 0 1 2 3 4 5 6

f(x) 0.30 0.20 0.20 0.10 0.10 0.05 0.05

F(x)

Mean/ Expected Value

Definition: themean or expected value of a discrete random variable X is E[X] = =

x

xf(x) (ie: sum over all possible values of x)

Interpretation: The expected value of a distribution is the balance point of thedistribution (similar to the mean of the data set).

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Note: if all the values have the same probability (die rolling), this becomes the commonaverage from ch3

Example: What is the expected value of the number of dots on the side-up face afterrolling a die?

Example 5.1 (Hockey, continued again)

Find E(X)

Example 5.2 (continued again)

What is the expected number of defective units?

Variance

Definition: The variance of a discrete RV X is

V ar(X) =

all x

(x E(X))2f(x)

orV ar(X) =

all x

x2f(x) [E(X)]2 =E(X2) [E(X)]2

Also recall standard deviation of X: =

V ar(X)

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Example 5.1 (Hockey, once more with feeling)

Find the variance and standard deviation for the Hockey data.

Example 5.2 (once more with feeling)

Compute the variance and standard deviation of the number of defective units.

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