01
Converting to standard normal
Let
, from which we know the PDF of . Show that is a standard normal random variable. In other words, verify that the PDF of
is . Hint: Use a method analogous to one in the lecture notes.
Solution
01
(1) Express
in terms of .
(2) Compute
.
(3) State formula for finding the PDF of
.
(4) State PDF for
.
(5) Compute PDF for
using the formula. Link to original
02
Symmetry of
Show that
for all .
Solution
01
(1) Express
in terms of .
(2) Compute
.
(3) State formula for finding the PDF of
.
(4) State PDF for
.
(5) Compute PDF for
using the formula. Link to original
03
Generalized normal
Let
be generalized normal variable with and . Using a chart of values, find: (a)
(b)
such that (c)
(Hint: Use and to avoid integration.)
Solution
03
(a)
(1) Write desired probability in terms of
values.
(2) Evaluate using a
value table.
(b)
(1) Use the
value table to find if
(2) Solve for
knowing that .
(c)
Recall formula for
and solve for Plug in
for and for and compute . Link to original
04
Normal distribution - test scores
In a large probability theory exam, the scores are normally distributed with a mean of 75 and a standard deviation of 10.
(a) What is the probability that a student scored between 70 and 80
(b) What is the lowest score a student can achieve to be in the top 5%?
(c) What score corresponds to the 25th percentile?
Solution
04
(a)
(1) Write desired probability in terms of
values.
(2) Use table to evaluate expression.
(b)
(1) Interpret problem.
Since we wish to find the top
, we wish to find such that .
(2) Use lookup table to find
.
(3) Given that
, solve for .
(c)
(1) Interpret problem.
Since we wish to find the 25th percentile, we wish to find
such that .
(2) Use lookup table to find
.
(3) Given that
, solve for . Link to original
05
Normal distribution - cars passing toll booth
The number of cars passing a toll booth on Wednesdays has a normal distribution
. (a) What is the probability that on a randomly chosen Wednesday, more than 1,400 cars pass the toll booth?
(b) What is the probability that between 1,000 and 1,400 cars pass the toll booth on a random Wednesday?
(c) Suppose it is learned that at least 1200 cars passed the toll booth last Wednesday. What is the probability that at least 1300 cars passed the toll booth that day?
Solution
14
(a)
(1) Write desired probability in terms of
values.
(2) Use lookup table to compute probability.
(b)
(1) Write desired probability in terms of
values.
(2) Use lookup table to compute probability.
(c)
(1) Set up conditional probability expression.
(2) Write desired probability in terms of
values.
(3) Use lookup table to compute probability.
Link to original
06
Gaussian basics
Find the probability that one observation of a Gaussian variable will yield a value within 1.5 standard deviations of the mean.
Solution
07
Morning commute time
My morning commute time is normally distributed, with a mean of 14 minutes, and a standard deviation of 4 minutes. I leave for work every morning at 8:45am and need to arrive by 9:00am.
(a) On any given day, what is the probability that I am late?
(b) On any given day, what is the probability that I reach before 8:55am?
Solution
08
(a)
(b)
Link to original
08
Blood sugar testing
A test for diabetes is a measurement
of a person’s blood sugar level following an overnight fast. If a person has diabetes, is a Gaussian random variable. Now suppose that the doctor has decided to use the following scores: positive for diabetes if
140, negative if , and the test is inconclusive if . Find the probability the test result will be inconclusive for a person who has diabetes.
Solution