Practice problems for Midterm 1
Problem 1
You are given two coins, one fair and the second one that
comes up heads with probability
.
7. Initially, you have no information
which coin is biased (the two coins look the same). You toss each of
the two coins twice, and coin A comes up heads more times than coin
B does. What is the probability that coin A is the biased one?
Problem 2
The pdf of a continuous random variable
X
is given by
f
X
(
x
) =
a
if 0
< x <
1
b
if

1
< x <
0
.
The random variable has the mean
EX
= 1
/
4.
(
a
) Find
a
and
b
and plot the density.
(
b
) Compute and plot the cumulative distribution function of
X
.
(
c
) Find
P
(
X
≤
1
/
2),
P
(
X
=

2
/
3) and
P
(
X
= 0).
Problem 3
A continuous random vector (
X, Y
) has a joint pdf given
by
f
X,Y
(
x, y
) =
6
y
if 0
< y <
1 and
y < x <
2

y
0
otherwise
.
(
a
) Find the marginal probability densities of
X
and
Y
. Are
X
and
Y
independent?
(
b
) Compute the probability
P
(
X
+
Y >
1).
Problem 4
A sack contains several boxes, each containing a die.
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 '10
 FROHMADER
 Probability theory, probability density function, marginal probability densities, continuous random vector

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