550.311 Probability & Statistics - FINAL EXAM SOLUTIONS
FALL 2012
1. An experiment is to independently toss a coin n times and record the number X of heads. The coin has
a probability of heads. We wish to use the results to test the simple null Ho : = 1 a
550.311 Probability & Statistics - SPRING 2011
MIDTERM 1 SOLUTIONS
1. Suppose X and Y are jointly continuous random variables with the following joint probability density
function:
2xey if 0 < x < 1, y > 0
f (x, y ) =
0
otherwise
(a) Compute P (0 < X < 1
550.311 Probability & Statistics - SPRING 2011
NAME
MIDTERM 1
SECTION
Directions: show all work in the space provided below each problem - use the back of the page if
necessary.
1. Suppose X and Y are jointly continuous random variables with the following
550.311 Probability & Statistics - Fall 2012
MIDTERM 2 solutions
1. SHORT ANSWERS.
(a) Suppose X has a normal distribution with mean and variance 2 . The moment-generating function of
1 22
X is MX (t) = et+ 2 t . Show that Z := X has a standard normal dis
550.311 Probability & Statistics - Fall 2012
NAME
MIDTERM 2
SECTION
Directions: show all work in the space provided below each problem - use the back of the page if
necessary.
1. SHORT ANSWERS.
(a) Suppose X has a normal distribution with mean and varianc
550.311 Probability & Statistics - Fall 2012
MIDTERM 1 solution
1. SHORT ANSWERS.
(a) A discrete random variable X has the following probability mass function: p(x) = (.2)(.8)x1 for x =
1, 2, 3, . . . . Compute P (X > 2).
P (X > 2) = 1 P (X = 1) P (X = 2)
FALL 2012
NAME mmmmmmm ....;. wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww __ SECTION ____________ __
Directions: show all work in the space provided below each problem use the back of the page if
necessary.
1. An experiment is to independently toss a
550.311 Probability & Statistics - FALL 2011
MIDTERM 1 solution
1. Suppose X and Y are jointly continuous random variables with the following joint probability density
1
function:
2 (x + 3y ) for 0 < x < 1, 0 < y < 1
f (x, y ) =
0
elsewhere
(a) Separately
550.311 Probability & Statistics - FALL 2011
NAME
MIDTERM 1
SECTION
Directions: show all work in the space provided below each problem - use the back of the page if
necessary.
1. Suppose X and Y are jointly continuous random variables with the following j
550.310 Probability & Statistics - Spring 2013
MIDTERM 2 solutions
cfw_
1. Let X and Y be jointly continuous rvs having joint p.d.f. f (x, y ) =
x + y for 0 < x < 1, 0 < y < 1
.
0
otherwise
(a) Compute E (XY ).
cfw_
11
11 2
x3 y
2 ) dx dy = 1
E (XY ) = 0
550.310 Probability & Statistics - SPRING 2013
MIDTERM 1 SOLUTION
1. Suppose X and Y are random variables that have the following joint distribution:
x y if 1 x 2, 0 y 1
f (x, y ) =
0
if otherwise.
(a) Determine the marginal distribution of X . If 1 x 2,