Probability Basic Probability Concepts Probability Distributions Sampling Distributions

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.