1. CS723 - Probability and Stochastic Processes

2. Lecture No. 11

3. In Previous Lectures
Discussion of discrete random variables and their associated concepts Thorough analysis of gambling games like chuck-a-luck and prize bonds Jointly random variables and their joint distribution function PMF and CDF Marginal distributions of two random variables from joint distributions Independence of two random variables
Use your best judgment

4. Conditional Distribution
The probabilities or random variables change due to additional information The collection of conditional probability values give conditional PMF/CDF Conditioning can be on any event
Use your best judgment

5. Conditional PMF
The graphics people should re-generate these graphs. The values are obvious and I talk about them in lecture. If the three graphs are to be shown on the same slide, they should appear one after the other as I speak about them. The three graphs should be drawn in such a way that grid lines (dotted) are properly aligned.

6. Expected Value
Average value of the random variable from a large collection of outcomes Example: a class with 100 students In general, E(X) = ∑ xi Pr(X=xi) = ∑ xi pi Chuck-a-luck is an unfair game because the expected value is Prize bonds have a expected value of for 1500 rupee prize bonds Are prize bonds a fair game? No!
This slide can benefit from a simple animation. Show silhouettes of students representing class. Be creative. Listen to what I say about this and generate an appropriate simple animation.
The last “No!” should appear a short while after “Are prize bonds a fair game?” is displayed. They should be synchronized with my words.

7. Expected 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 distribution is always λ The value of λ chosen for Poisson approximation was 24
Use your best judgement

8. Geometric Interpretation
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

9. Transformation of RV’s
Random variable maps outcome to point on real line Other mappings can be introduced before assignment of probabilities Additional mappings can involve one random variable R→R, two random variables R2 → R, or more RV’s Examples: Z1 = X+Y , Z2 = |X-Y| Y = 2X , Y = X – 3 , Y = 2X + 3Y

10. Expected Value of Transformed RV
If Y is a random variable obtained from random variable X using Y = g(X) E(Y) = ∑ yi Pr( Y=yi ) = ∑ yi pyi = ∑ g(Xi) Pr( X=xi ) = ∑ g( xi ) pxi Works for 1→1 mappings as well as for many → 1 mappings
There should be an ‘i’ below the summation sign ∑.

11. 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+1 E(Y) = (75*2 + 15*3 + 1*4)/216 = 0.92 For prize bonds with inflation effect Y = X – 30 and E(Y) = E(X) – 30 = E(X+a) = E(X) + a & E(bX) = bE(X)
Use your best judgement.

12. Moments
Expected values of transformed RV’s of the kind Y = Xp and Y = (X – m)p The first type are simple moments and second type are central moments Expected value of Y = (X – m)2 is called variance of random variable X
Use your best judgment.