Due date: Thursday 10/2, 11:59pm
Normal distribution
01
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.
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02
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?
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08
(a)
(b)
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03
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?
Link to originalSolution
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
04
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.) Link to originalSolution
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
Review problems
05
02
Insurance expected payout
A car insurance analytics team estimates that the cost of repairs per accident is uniformly distributed between $100 and $1500. The manager wants to offer customers a policy that has a $500 deductible and covers all costs above the deductible.
How much is the expected payout per accident?
(Hint: Graph the PDF for the cost of repairs
; write a formula for the payout in terms of using cases; then integrate.) Link to originalSolution
06
(1) Find PDF of
. If
, insurance covers 0$. If
, then the insurance covers dollars.
(2) Integrate to find
. Since the cost of repairs in uniformly distributed, we have
, . Link to original
06
05
CDF of derived variable
Suppose
is a continuous random variable. Let . Find the CDF of . Link to originalSolution
09
Case 1:
Case 2:
Therefore:
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07
06
Octane revenue
The owner of a small gas station has his 1,500 gallon tank of 93-octane gas filled up once at the beginning of each week. The random variable
is the amount of 93-octane the station sells in one week (in thousands of gallons). The PDF of X is shown below. Assuming the station consistently charges $3.00 per gallon for 93-octane and pays $2,000 for the weekly fill-up, find the CDF of
, the profit the station makes from the 93-octane in a week. Link to originalSolution
10
Case 1:
Case 2:
Therefore:
Link to original