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Probability Basic Probability Concepts Probability Distributions Sampling Distributions

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Probability

Basic Probability Concepts Probability Distributions Sampling Distributions

Probability

Basic Probability Concepts

Basic Probability Concepts

Probability refers to the relative chance that

an event will occur. It represents a means

to measure and quantify uncertainty.

0 probability 1

Basic Probability Concepts

The Classical Interpretation of Probability:

P(event) = # of outcomes in the event

# of outcomes in sample space

Example:

P(selecting a red card from deck of cards) ?

Sample Space, S = all cards Event, E = red card

thenP(E) = # outcomes in E = 26 = 1

# outcomes in S 52 2

Probability

Random Variables and Probability

Distributions

Random Variable

A variable that varies in value by chance

Random Variables

Discrete variable - takes on a finite, countable # of values

Continuous variable - takes on an infinite # of values

Probability Distribution

A listing of all possible values of the random variable, together with their associated probabilities.

Notation:

Let X = defined random variable of interest x = possible values of X P(X=x) = probability that takes the value x

Example:

Experiment:

Toss a coin 2 times.

Of interest: # of heads that show

Example:

Let X = # of heads in 2 tosses of a coin (discrete)

The probability distribution of X, presented in tabular form, is:

x P(X=x) 0 .25 1 .50 2 .25

1.00

Methods for Establishing Probabilities

Empirical Method Subjective Method Theoretical Method

Example:

Toss 1 Toss 2

T T There are 4 possible

T H outcomes in the

H T sample space in this

H H experiment

Example:

Toss 1 Toss 2

T T P(X=0) = ?

T H Let E = 0 heads in 2 tosses

H T P(E) = # outcomes in E

H H # outcomes in S

= 1/4

Example:

Toss 1 Toss 2

T T P(X=1) = ?

T H Let E = 1 head in 2 tosses

H T P(E) = # outcomes in E

H H # outcomes in S

= 2/4

Example:

Toss 1 Toss 2

T T P(X=2) = ?

T H Let E = 2 heads in 2 tosses

H T P(E) = # outcomes in E

H H # outcomes in S

= 1/4

Example:Example:

The probability distribution in tabular form:

x P(X=x) 0 .25 1 .50 2 .25

1.00

Example:Example:

The probability distribution in graphical form:

P(X=x)1.00

.75

.50

.25

0 1 2 x

Probability distribution, numerical summary form:

Measure of Central Tendency:mean = expected value

Measures of Dispersion:variancestandard deviation

Numerical Summary Measures

Expected Value

Let = E(X) = mean = expected value

then

= E(X) = x P(X=x)

Example:

x P(X=x)

0 .25 1 .50 2 .25

1.00

= E(X) = 0(.25) + 1(.50) + 2(.25) = 1

Variance

Let ² = variance

then

² = (x - )² P(X=x)

Standard Deviation

Let = standard deviation

then = ²

Example:

x P(X=x)

0 .25 1 .50 2 .25

1.00

² = (0-1)²(.25) + (1-1)²(.50) + (2-1)²(.25)

= .5

= .5 = .707

Practical Application

Risk Assessment:

Investment A Investment B

E(X) E(X)

Choice of investment – the investment that yields the highest expected return and the lowest risk.