Summations
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?
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?
03
Counting flip flops
A bag contains 50 marbles, 30 blue and 20 red. A sequence of zeros and ones is created by pulling the marbles out one at a time (without replacement) and writing a 1 if the marble drawn is blue and a zero if it is red.
How many pairs of adjacent digits in the sequence are expected to differ from each other?
Hint: Use a sum of 49 indicators.
Central Limit Theorem
04
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.
05
Continuity correction: De Moivre-Laplace
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.
06
Normal approximation - Ventilator filters
A mechanical ventilator model uses air filters that last 100 hours on average with a standard deviation of 30 hours.
How many filters should be stocked so that the supply lasts 2,000 hours with probability at least 95%? Use a normal approximation to estimate the answer.
State the reason that the normal approximation is applicable.
07
Normal 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 minutes. 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.
08
Indicator method, exchangeability, summation rules
A class has 40 students: 24 women and 16 men. Each period the teacher selects a random student to present an exercise on the board from among those who have not presented already.
Let
count the number of times a man was chosen after 15 class periods. (a)
Find . (b) Find . Hint: Is
independent of ? Do you know anyway?
09
Graphing convergence to a bell curve Let
be independent RVs each having the following PMF:
Notice that
and for each . Define
. So and , and therefore . By the CLT,
should converge to the standard normal distribution as . In this problem you explore the limit process by direct computation of the cases . (a) Compute the PMF of
in terms of . Hint steps:
is the number of sequences of outcomes of having a total sum of , times the probability of any particular such sequence. - Find the probability of any particular sequence of
outcomes of . - There are
ways to get outcomes of and outcomes of . Solve for in terms of . - Put 2. and 3. together in 1. to get your formula.
(b) Compute the PMF of
for using your formula from (a) together with a scaling (derived variable calculation). (c) Draw the graphs of the PMF of
and the PMF of for .