ORIE 3500/5500 Engineering Probability and Statistics II Fall 2010 Assignment 2 Problem 1 A set of k coupons, each of which is independently a type j coupon with probability pj , n=1 pj = 1, is collected. Find the j probability that the set contains a typ

ORIE 3500/5500 Engineering Probability and Statistics II
Fall 2013
Assignment 10
Problem 1 The breaking strength of a certain type of cloth is to be
measured for 20 specimens. The underlying distribution is normal with
unknown mean but with a standard dev

ORIE 3500/5500 Engineering Probability and Statistics II
Fall 2013
Assignment 11
Problem 1 A bowl contains 20 chips, r of which are red, and the
remainder black. To test
H0 : r = 10 vs. H1 : r = 15,
3 chips are selected at random without replacement from

ORIE 3500/5500 Engineering Probability and Statistics II
Fall 2013
Assignment 2
Problem 1
is given by
The distribution function (cdf) of a random variable X
1
if x < 0
2 ex
3
if 0 x < 5
4
FX (x) =
.
1
x
12 + 3 if 5 x < 6
1
if x 6
(a) Plot this cdf. Is X

Further properties of the bivariate Gaussian distribution
Let X and Y be jointly normal distribution with 2 2 the means X and Y , variances X and Y , and correlation .
Suppose that the correlation = 0. Then the joint pdf fX,Y (x, y ) = fX (x)fY (y ) and

Properties of the bivariate normal distribution:
Let (X, Y ) be jointly normal with the means X 2 2 and Y , variances X and Y , and correlation .
Marginal distributions of X and Y :
2 X N ( X , X )
and
2 Y N (Y , Y ).
Conditional distributions: given X

The Law of Total Probability in the context of random variables
Let X1, X2, . . . , Xn be random variables;
some of the Xis may be discrete, and some other continuous;
let A be some event expressed in terms of X1, . . . , Xn.
The law of total probabili

Example: The number of claims arriving to an insurance company in a week is a Poisson random variable N with mean 20. Assume that the amounts of dierent claims are independent exponentially distributed random variables with mean 800.
Assume that the claim

Example: Let (X, Y ) be a continuous random vector with a joint pdf fX,Y (x, y ) = 15xy 2, 0 x, y 1, y x.
Compute the conditional densities fX |Y and fY | X ; Compute the conditional mean and the conditional variance of Y given X = x for 0 < x < 1.
Compu

Terminology:
the signicance level of the test: an upper bound on the probability of type 1 error. It is also often denoted by .
typical values of the signicance level: 0.1, 0.05, 0.01, 0.005.
The signicance level of the test is also called the size of

The choice of the prior distribution on the unknown parameter is usually done to reect our prior beliefs about the unknown parameter. For example, with the normal prior, N (, 2), we believe that the unknown parameter is, about, , and the less condent we

ORIE 3500/5500 Engineering Probability and Statistics II Fall 2010 Assignment 4 Problem 1 The joint pmf of X and Y is given by 1 pX,Y (i, j ) = for 1 i, j n, |i j | 1, 3n 2 where a positive integer n 4 is a parameter. (a) Verify that this is a legitimate

ORIE 3500/5500 Engineering Probability and Statistics II Fall 2010 Assignment 3 Problem 1 This exercise will be easy for those familiar with the Japanese puzzles called nonograms. The marginal pmfs of discrete random variables X and Y are given in the fol

ORIE 3500/5500 Engineering Probability and Statistics II Fall 2010 Assignment 1 Problem 1 We toss a coin 3 times. For this experiment we choose the sample space = H HH, T HH, HT H, HHT, T T H, T HT, HT T, T T T , where T stands for tails and H stands for