Download Practice Second Test

AMS 311
Practice Test 2 A
Spring 2006
Prof. Tucker
1. What is the probability that a TV will die in the next 2000 hours of use if it has already been used for
5000 hours and the length of its life has an exponential distribution with mean lifetime of 6000. (To
make the calculation easier, measure time in 1000’s of hours.)
2. An experiment is run 300 times with chances of success of 1/4. Use a normal approximation to
estimate the probability of having at most 60 successes. Express the answer in terms of the distribution
function Φ(x) of a standard normal.
3. Let X, Y be independent random variables with density function f(x,y) = 8xy, 0<x<y<1.
a) Find the marginal density f(x) of X and then write an integral to find E(X).
b) Find f(x|y) and write an integral to find E(X|y).
4. Let X and Y be independent random variables with density function 4e-2x e-2y, 0<x,y<∞.
Pr(X+Y < a) and from this determine the density function for X+Y, fX+Y(a).
Determine
5. Let X1, X2, X3, X4 be independent uniform [0,1] random variables. Set up an integral to determine
the probability that the smallest value of these random variables is less than 1/4.
6. Let X and Y have the density function e-(x+y), 0<x,y<∞. Let W = X+Y, and Z = X/(X+Y).
a) Do a change of variables from X,Y to W,Z to obtain the density function for W,Z.
b) Find the marginal density function for Z.
(Hint
∫ ue
−u
= −ue− u + ∫ e− u )
AMS 311
Practice Test 2 B
Spring 2006
Prof. Tucker
1. What is the probability that a computer will die in the next 3000 hours of use if it has already been
used for 8000 hours and the length of its life has an exponential distribution with mean lifetime of 8000.
(To make the calculation easier, measure time in 1000’s of hours.)
2. An experiment is run 10000 times with chances of success of 1/2. Use a normal approximation to
estimate the probability of having at least 5,050 successes. Express the answer in terms of the
distribution function Φ(x) of a standard normal.
3. Let X, Y be independent random variables with joint density function f(x,y) = 24y(1-x-y), 0 < x, 0 <y,
x+y< 1.
a) Find the marginal density f(x) of X and then write an integral to find E(X).
b) Find f(x|y) and write an integral to find E(X|y).
4. Let X and Y be random variables with density function 4xy, 0<x,y<1 Determine Pr(X+Y < a).
Warning: it makes a big difference whether 0 < a ≤ 1 or 1 < a < 2. (this problem is more complex than a
real test question.)
5. Let X1, X2, X3, X4, X5 be independent uniform [0,1] random variables. Set up an integral to
determine the probability that the median value of these random variables is greater than 1/2.
6. Let X and Y have the density function e-(x+y), 0<x<∞. Let W = X/Y, and Z = Y.
a) Do a change of variables from X,Y to W,Z to obtain the density function for W,Z.
b) Write an integral for finding the marginal density function for W.