Math 394 B&C. July 11, 2014
Problem 1. We need to choose a committee of six people: three French and three Germans,
out of six French and seven Germans. How many ways are there to do this? Your answer should
be in the form of a number, say 10 or 23
Lecture 17. Central Limit Theorem I
August 4, 2014
Suppose X1 , X2 , X3 , . . . are iid (independent identically distributed) random variables with
mean EX1 = EX2 = . . . = and variance Var X1 = Var X2 = . . . = 2 . Consider the sum
SN := X1 + . . . + XN
Lecture 19. Continuous Distributions I
August 8, 2014
A random variable X has continuous distribution with density p(x) if for every a and b
P(a X b) =
Unlike discrete random variables, this one takes any xed value with probability zero: P(X
Lecture 18. Central Limit Theorem II
August 6, 2014
Equity Buer, or Value at Risk. An insurance company has N clients. The kth client
has insurance claim Xk . Suppose X1 , X2 , X3 , . . . are iid (independent identically distributed)
random variables with
Lecture 3. Combinatorial Story Problems
June 27, 2014
Problem 1. Five women and four men take an exam. We rank them from top to bottom,
according to their performance. There are no ties.
(i) How many possible rankings?
(ii) What if we rank men and women s
Lecture 4. Multinomial Coecients
June 30, 2014
Problem 1. Suppose we need to put seven numbers: 1, 2, 3, 4, 5, 6, 7 into three bags: a
green bag, a yellow and a blue one. We need to put three numbers into the green bag, two
numbers in the yellow bag and t
Lecture 6. Inclusion-Exclusion Principle
July 7, 2014
Inclusion-Exclusion Formula for Three Dimensions
P(A B C) = P(A) + P(B) + P(C) P(A B) P(A C) P(B C) + P(A B C)
Indeed, let p1 be the probability of the event 1 on the diagram, which is A \ (B C). Let p
Lecture 5. Axioms of Probability
July 2, 2014
The following set of axioms was formulated by a Russian mathematician Andrey Kolmogorov. We have the set of elementary outcomes , and subsets A are called
events. Each event A has a number P(A), which is calle
Lecture 2. Properties of Combinations
June 25, 2014
Symmetry. We can say without calculations that 8 = 8 . Indeed, for every subset of
cfw_1, 2, . . . , 8 of two elements there is a subset of six elements: its complement. For example,
cfw_3, 5 corresp
Lecture 1. Permutations and Combinations
June 23, 2014
A permutation of 123 is, say, 321 or 231: numbers cannot repeat. There are 3 2 1 = 3! = 6
permutations: there are 3 chioces for the rst slot, 2 choices for the second slot (because one
of the numbers
Lecture 8. Second Bayes Formula
July 11, 2014
Recall from the last lecture: we have events F1 , . . . , Fn such that
F1 F2 . . . Fn = , Fi Fj = , i = j.
In plain English, this means that one and only one of the events F1 , . . . , Fn happens. Take an
Lecture 7. Conditional Probability
July 9, 2014
A bag with 11 cubes:
3 red and fuzzy
2 red and smooth
4 blue and fuzzy
2 blue and smooth
Put your hand in the bag and randomly pick a cube. Let R = cfw_it is red, F = cfw_it is fuzzy.
Then the probabilit
Lecture 9. Independent Events
July 14, 2014
Events A and B are called independent if knowledge of whether A happened or not does not
inuence the probability of B:
P(B | A) = P(B).
P(B | A) =
we can rewrite this as
P(A B) = P(A)P(B)
Lecture 15. Binomial and Poisson Distributions
July 27, 2014
Binomial Distribution. What is X
Bin(N, p)? Toss a biased coin (with probability
p of Heads, and probability q = 1 p of Tails) N times. Equivalently, conduct N Bernoulli
trials so that each tria
Lecture 14. Poisson Distribution
July 25, 2014
Fix a parameter > 0. Consider a distribution on 0, 1, 2, 3, . . ., given by the formula
P(X = k) =
e , k = 0, 1, 2, . . .
This is called the Poisson distribution with parameter . We write: X
P(X = 0) =
Lecture 16. Geometric and Negative Binomial Distributions
August 1, 2014
Geometric Distribution. Toss a biased coin, with probability p of Heads, and probability
q = 1 p of Tails. How many X tosses do you need to get your rst Heads? This is called geometr
Math 394 B&C. Probability I. Summer 2014. Homework 3, due July 16
Homework 3 Solution, due July 16
Problems from old actuarial exams are marked by a star.
Problem 1*. Upon arrival at a hospital emergency room, patients are categorized according to their c
Math 394 B&C. Probability I. Summer 2014. Homework 1, due July 2
Homework 1 Solution, due July 2
Problem 1. There are 8 apartments for 6 people. Each person chooses one apartment, and
each apartment can host no more than one person. How many
Math 394 B&C. Probability I. Summer 2014. Homework 4, due July 23
Homework 4 Solution, due July 23
Problem 1. Let X be the random number on a die: from 1 to 6.
(i) What is the distribution of X?
(ii) Calculate EX.
(iii) Calculate EX 2 .
Math 394 B&C. Probability I. Summer 2014. Homework 5, due July 30
Homework 5 Solution, due July 30
Binomial and Poisson Distributions
Problem 1. Explain why Var X = for X
Solution. Var X = EX 2 (EX)2 , and
P(X = k) =
e , k = 0, 1, 2, . . .
Math 394 B&C. Probability I. Summer 2014. Homework 6, due August 6
Homework 6 Solution
Problem 1. (i) Find P(X 2) for X Geo(1/2).
(ii) Find P(X 5) for X NB(3, 1/3).
(iii) Find P(X 3) for X NB(3, 1/3).
(iv) Find P(X = 0) for X Geo(1/4).
Solution. (i) P(X 2
Quiz 2 Solution
Math 394 B&C. July 18, 2014
Problem 1. You have three coins, two of them fair and the third a magic one, which rolls out
Heads with probability 75%. Suppose you picked randomly one coin and tossed it once. You got
Heads. What is the probab
Quiz 4 Solution
Math 394 B&C. August 1, 2014
Problem 1. You take ve preliminary exams, and you will pass each exam with probability
1/3, independently of other exams. What is the probability that you will pass two or more exams?
Solution. The number X of
Math 394 B&C. August 8, 2014
Problem 1. There are N = 100 clients, each will experience ood with probability p = 0.5%,
independently of others. What is the probability that exactly one client will experience ood?
(Hint: use Poisson approximation.)
Quiz 3 Solution
Math 394 B&C. July 25, 2014
Problem 1. Toss a fair coin twice and let X be the number of Heads, and let Y = 1 if the
rst toss is Tails, Y = 0 otherwise. Find: (i) the distribution of X; (ii) the distribution of Y ; (iii)
EX; (iv) EY ; (v)
Lecture 11. Independent Random Variables
July 18, 2014
Recall that a random variable is a function X : R. For example, toss a coin twice.
Then = cfw_HH, HT, T H, T T . Let X be the number of tails. Then
X(HH) = 0, X(T H) = X(HT ) = 1, X(T T ) = 2.
Lecture 12. Variance and Covariance
July 21, 2014
Covariance. Random variables X and Y have covariance:
Cov(X, Y ) = E(X EX)(Y EY ).
This is a measure of the following: when one variable increases, does the other increase or
decrease (on average)? We can
Lecture 13. Binomial Distribution
July 23, 2014
Suppose you have a biased coin, which falls H with probability p, 0 < p < 1. You toss it N
times. Let X be the number of H. What is the distribution of the random variable X? We call
this the sequence of Ber
Lecture 10. Random Variables
July 16, 2014
A random variable is a function X : R. For example, toss a coin twice. Then
= cfw_HH, HT, T H, T T . Let X be the number of tails. Then
Y (HH) = 0, Y (T H) = Y (HT ) = 1, Y (T T ) = 2.
The distribution of a rand