CS723 - Probability
and
Stochastic Processes
CS723 - Probability
and
Stochastic Processes
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Lecture 11: Conditional Probability, Expected Value, and Random Variable Transformations, Slides of Probability and Stochastic Processes
A part of the lecture notes for cs723 - probability and stochastic processes, specifically lecture no. 11. The lecture covers conditional probability, expected value, and transformations of random variables. Topics include conditional pmf/cdf, geometric interpretation of expected value, and transformation of random variables. Examples of expected value calculations for bernoulli and poisson distributions are provided, as well as the concept of fairness in gambling games like chuck-a-luck and prize bonds.
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2011/2012
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CS723 - Probability
and
Stochastic Processes
CS723 - Probability
and
Stochastic Processes
- Lecture No. 11Lecture No.
Conditional DistributionConditional DistributionThe probabilities or random variables
change due to additional information The collection of conditional probability
values give conditional PMF/CDF The probabilities or random variablesConditioning can be on any event
change due to additional information The collection of conditional probability
values give conditional PMF/CDF Conditioning can be on any event
Conditional PMFConditional PMF
0.1 0.1 0.
Expected ValueExpected Value
Expected value of Bernoulli distribution
B(N,p) is Np Expected number of customers going
into a restaurant in 10 minutes was 24 Expected value of Poisson distributionis^
always
λ
The value of
λ^
chosen for Poisson
approximation was 24 Expected value of Bernoulli distribution
B(N,p) is Np Expected number of customers going
into a restaurant in 10 minutes was 24 Expected value of Poisson distributionis^
always
λ
The value of
λ^
chosen for Poisson
approximation was 24
Geometric InterpretationGeometric InterpretationExpected value of a random variable is
inner product of two vectors The result could be interpreted as net
torque of a rod with hanging weights Expected value of a random variable is
inner product of two vectors The result could be interpreted as net
torque of a rod with hanging weights
Expected Value ofExpected Value ofTransformed RVTransformed RV
If Y is a random variable obtained from
random variable X using Y = g(X) E(Y) =
∑^
y^ Pr( Y=yi^
) =i^
∑^
y^ pi^
yi
=^ ∑
g(X
) Pr( X=xi^
) =i^
∑^
g( x
) pi^
xi
Works for 1
1 mappings as well as for
many
1 mappings
If Y is a random variable obtained from
random variable X using Y = g(X) E(Y) =
∑^
y^ Pr( Y=yi^
) =i^
∑^
y^ pi^
yi
=^ ∑
g(X
) Pr( X=xi^
) =i^
∑^
g( x
) pi^
xi
Works for 1
1 mappings as well as for
many
1 mappings
Expectation of Y=g(X)Expectation of Y=g(X) Gambling game of chuck-a-luck run by a
benevolent gambling house You don’t loose your bet and get a
chance to win 4,3, or 2 dollars. Transformed RV is Y = X+1E(Y) = (752 + 153 + 1*4)/216 = 0.92For prize bonds with inflation effectY = X – 30 and E(Y) = E(X) – 30 = -5.56E(X+a) = E(X) + a
& E(bX) = bE(X)
Gambling game of chuck-a-luck run by a
benevolent gambling house You don’t loose your bet and get a
chance to win 4,3, or 2 dollars. Transformed RV is Y = X+1E(Y) = (752 + 153 + 1*4)/216 = 0.92For prize bonds with inflation effectY = X – 30 and E(Y) = E(X) – 30 = -5.56E(X+a) = E(X) + a
& E(bX) = bE(X)