Need Help-Statistics Assignment


Need Help-Statistics Assignment
Statistical Inference I: J. Lee
Assignment 3
Problem 1.
Approximately 80,000 marriages took place in the state of Pennsylvania last year. Estimate
the probability that, for at least one of the couples married in PA last year,
(a) both partners were born on April 30;
(b) both partners celebrated their birthday on the same day of the year.
Make sure to state what assumptions you are making in your computation.
Problem 2.
A certain typing agency employs Al, Bob, and Cathy as typists. The average number of errors
per page is 3 when typed by Al, 4.2 when typed by Bob, and 2.1 when typed by Cathy. If your 7-page article
is equally likely to be typed by any of the three typists, estimate the probability that it will have no errors.
Also estimate the probability it will have at most 3 errors.
Problem 3.
Let
X
denote the lifetime (in hours) of a light bulb, and assume that the density function of
X
is given by
f
(
x
) =
8
<
:
2
x
if 0
x <
1
=
2
3
=
4
if 2
< x <
3
0
otherwise.
(a) On average, what fraction of light bulbs last more than 15 minutes?
(b) Compute
E
(
X
).
(c) Compute
P
(0
:
25
< X
2
:
2
j
X >
1)
:
(d) Compute
P
(
X
= 2)
; P
(
X
= 0)
; P
(
X
=
E
(
X
))
:
Problem 4.
The density function of
X
is given by
f
(
x
) =
a
+
bx
2
if 0
x
1
0
otherwise.
Suppose also that you are told that
E
(
X
) = 3
=
5
:
(a) Find
a
and
b
.
(b) Determine the cdf,
F
(
x
), explicilty.
Problem 5.
De ne the function
F
:
R!R
by
F
(
x
) =
8
>
>
<
>
>
:
0
if
x <
0
x=
2
if 0
x <
1
(
x
+ 2)
=
6
if 1
x <
4
1
if
x
4
:
(a) Is
F
the cdf of a continuous random variable? Explain your answer.
(b) If your answer to part (a) is \yes”, determine the corresponding pdf; if your answer was \no”, then
make a modi cation to
F
so that it is a cdf, and then compute the corresponding pdf.
(c) Compute the expectation of
X
, if
X
has density given by the pdf from part (b).
1
Need Help-Statistics Assignment
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