Probability Basic Probability Concepts Probability Distributions Sampling Distributions.

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Transcript of 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 = (01)²(.25) + (11)²(.50) + (21)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.