{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MA519_JAN95

# MA519_JAN95 - 100 be independent variables each with a...

This preview shows pages 1–2. Sign up to view the full content.

QUALIFYING EXAMINATION JANUARY 1995 MATH 519 25 points/problem 1. Let X 1 , X 2 , . . ., X 11 be independent identically distributed random variables each having density f ( t ) = e - t , t > 0, f ( t ) = 0 elsewhere. Find the densities of the maximum of the eleven X i s, of the minimum of the eleven X i s, and of the second largest of the eleven X i s. 2. Let ( X, Y ) be uniformly distributed with on the circumference of the unit square { ( x, y ) : | x | = 1 or | y | = 1 } . (So, for example, P ( a < X < b and Y = 1) = ( b - a ) / 8 if - 1 a < b 1). Find the distributions of X + Y and XY . Are X and Y independent? Find the joint density of Θ( X, Y ), where Θ is a random rotation of a point about (0 , 0) through an angle uniformly distributed on [0 , 2 π ]. 3. Let X 1 , . . ., X 100 be independent normal (0 , 1) variables. Let Y 1 , . . ., Y 100 be in- dependent variables each with distribution P ( Y i = 1) = P ( Y i = - 1) = 1 2 . Let

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 100 be independent variables each with a uniform distribution on (-1 , 1). Suppose all the 300 X s, Y s, Z s are independent. i) Is 100 X i =1 X i exactly normal? If not, why not? ii) Is 100 X i =1 Y i exactly normal? If not, why not? iii) Is 100 X i =1 Z i exactly normal? If not, why not? iv) Estimate P ( 100 X i =1 X i Y i Z i > 10), in terms of the standard normal distribution function. v) Find P ( 100 Y i =1 X i 100 Y i =1 Y i 100 Y i =1 Z i > 0). 4. A die is rolled until two sixes in a row come up. Find the expected number of rolls and the expected total of all the numbers rolled. Also ﬁnd the distribution of the number of sixes rolled and the distribution of the number of ﬁves rolled....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

MA519_JAN95 - 100 be independent variables each with a...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online