ECSE-2500, Engineering Probability, Spring 2010, Rensselaer Polytechnic Institute

# Lecture 17

# Review (page 29).

- Random experiment, e.g., voltage at wall.
- Sample space S of all possible outcomes.
- Types of sample spaces:
- Finite, e.g., which head of a coin is up
- Infinite discrete (countable), e.g., how many tosses until a head shows.
- Infinite continuous (uncountable), e.g., time until a uranium atom decays
- Combos are possible, but not in this course.

- Events are subsets of S.
- Event class F of interesting events. For discrete (finite or infinite) S, F can be all subsets.
- For continuous, that doesn't work for technical reasons. So, can use countable numbers of unions, complements of intervals.
- Probability is a real number assigned to each event.
- There are rules for what are legal probabilities.
- E.g., >=0, <=1, for disjoint events, probabilities sum.

# Pairs of random variables (Chapter 5, page 233).

- joint behavior: pmf, pdf, cmf (pmf is for discrete, pdf for continuous)
- summarize behavior: joint moments
- independent?, correlation
- conditional probabilities

# 1 experiment -> 2 random variables

- Get 2 random variables from the one experiment
- Random experiment: pick a file on your computer.
- 2 random vars: # whole blocks used and fraction of last partial block.
- Random experiment: pick a student in class
- 2 random vars: height, weight
- experiment may be defined broadly:
- Random experiment: toss 2 coins
- 2 random vars: face showing on 1st coin, face showing on 2nd coin.

# pmf, cdf, marginal probabilities for 2 coins

- do in class for fair coins
- do for unfair coins, 1st: p(H)=.6, 2nd: p(H)=.2.
- that was too easy because they really were 2 separate experiments.
- iclicker, what is p[both coins are heads]?
- A: .04
- B: .12
- C: .25
- D: .5
- E: .36

# level, cost of textbooks

# books | level | ||

cost range | 100 | 200 | 400 |

$50-100 | 4 | 3 | 0 |

$101-150 | 1 | 3 | 5 |

- compute joint pmf, joint cdf, marginal cdf, marginal probabilities
- iclicker, what is p[cost <=$100]?
- A: 0
- B: 1/2
- C: 7/16
- D: 4/5
- E: 1

- Find P[cost<=$100 and level > 2]]

# Continuous random variables - point in square

- find pdf, cdf, marginals
- P[X>Y]?

# Continuous random variables - point in triangle

- Triangle (0,0), (1,0), (0,1) (variant of exercise 5.16)
- experiment: pick a point uniformly distributed
- random variables: X,Y
- intuition: they're correlated since large X means likely small Y.
- pdf: find scale factor
- find cdf by integrating pdf
- P[X>Y]?

# Independence

- joint pmf/pdf is product of marginal pmfs/pdfs
- equivalently joint cdfs is product of marginal cdfs
- Are random variables in 2 coin example independent?
- ... textbook cost ...?
- ... square ...?
- ... triangle ...?

# Section 5.6

- Find mean, variance of 2 variables in each of 4 previous examples.
- joint moment
- central moment
- covariance {$ COV(X,Y) \overset{\Delta}{=} E\left[(X-E[X])(Y-E[Y])\right] = E[XY] - E[X]E[Y] $}
- independent -> COV=0; reverse not always true for nonlinear dependence.