Due date: Friday 11/07, 9:00am
Summations
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01
Summation of three: Rolling mixed dice
You have three dice. One has 4, one has 6, and one has 12 sides.
How many 4s do you expect to see if you roll these dice together?
Link to originalSolution
01
(1) Call the
-sided, -sided, and -sided dice, dice respectively. Let be the event that dice rolls a . Then
, , and .
(2) Let
be the number of ‘s that are rolled. Then . Thus, by linearity of expectation.
Thus, we expect to see
Link to original‘s if we roll these dice together.
02
02
Jumble of coins
In my pocket I have a jumble of coins: 5 dimes, 4 quarters, 3 nickels, 3 pennies, and one big 50
-piece. I draw three at random. What is the expected value of the three? Link to originalSolution
02
(1) Let
be the value of the first coin drawn, let be the value of the second coin drawn, and let be the value of the third coin drawn. The central trick to efficiently solve this problem is to notice that are all identically distributed. One can see this by the following argument: using an ordered triple
, write down all possible permutations of drawings. Notice that the number of triples where is a dime is equal to the number of triples where is a dime is equal to the number of triples where is a dime. We can further extend this observation to all the values. Thus, the distributions of
are all the same.
(2) Another, nicer, argument is to notice that we can swap
and in these ordered triples without changing the overall set, and similarly for and there exists a bijection between ; ; and identical distribution. Thus, we have that .
(3) Let
be the sum of the values of the three coins. Then, Now,
Thus,
Link to original.
Central Limit Theorem
03
09
Burning through light bulbs
A 100 Watt light bulb’s expected lifetime is 600 hours, with variance 360,000. An advertising board uses one of these light bulbs at a time, and when one burns out, it is immediately replaced with another. (The lifetime of each bulb is independent from the others.) Let the continuous random variable
be the total number of hours of advertising from 10 bulbs. (a) Find the expected value of
. (b) Find the variance of
. (c) Use the CLT to approximate the probability that
is less than 5,500 hours. (You should decide whether it is appropriate to use the continuity correction.) Link to originalSolution
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(a)
(b)
(c)
The standardized variable is:
Then:
Link to original
04
02
Link to originalDe Moivre-Laplace Continuity Correction
A fair die is rolled 300 times.
Use a normal approximation to estimate the probability that exactly 100 outcomes are either 3 or 6.
Do this with and without the continuity correction.
05
01
Normal approximation - Eating hot dogs
Frank is a competitive hot dog eater. He eats
in with . What is the probability that Frank manages to consume
in or less, in an upcoming competition? Use a normal approximation from the CLT to estimate this probability. State the reason that the normal approximation is applicable.
Link to originalSolution
04
(1) The normal approximation in this case is applicable since:
Assumptions:
- Frank eats a large number of hot dogs
the sample size, or , is sufficiently large - We assume that the amount of time Frank spends on each hot dog does not depend on how many he has had previously
the times to consume each hot dog are independent and identically distributed
(2) Let
be the time taken to eat the -th hot dog. Let be the time taken to eat hot dogs. Then seconds with seconds. Since
minutes is seconds, by the CLT we have: Link to original
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04
Link to originalNormal approximation - Grading many exams
An instructor has 50 exams to grade. The grading time for each exam follows a distribution with an average of 20 minutes and variance of 16. Assume the grading times per exam are independent.
Roughly what are the odds that after 450 minutes of grading, at least half the exams will be graded? Use a normal approximation to estimate the answer.
State the reason that the normal approximation is applicable.