Due date: Tuesday 11/11, 11:59pm
Summations
01
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.
Link to originalSolution
03
(1) Let
be a sequence of indicators where is the event that the -th entry differs from the -th entry. By a similar argument to above, the are identically distributed for each , and are, in fact, independent.
(2) Let
be the number of pairs of entries that differ from each other. Then . By the above,
(3) Let
denote the -th entry. Now, Thus,
Link to original.
Central Limit Theorem
02
11
Community college ages
At a community college, the mean age of the students is 22.3 years, and the standard deviation is 4 years. A random sample of 64 students is drawn.
What is the probability that the average age of the students in the random sample is less than 23 years?
Link to originalSolution
11
Sample mean RV:
has and . Standardize this and approximate with a normal distribution:
Link to original
03
03
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.
Link to originalSolution
06
In this case the normal approximation is applicable since we have a large sample size (need a large number of filters to last
hours) and they follow follow independent but identical distributions. Let
be how long the -th filter lasts. Let where we want to find such that . By normal approximation and the Central Limit Theorem, we have and thus
Link to originalfilters are required.
04
12
Winning the lottery
Suppose a lottery game requires that you purchase a $10 game card and advertises a 10% probability of winning a prize.
Use the Central Limit Theorem and the continuity correction to approximate the probability of winning at least 20 times when you purchase 100 of these game cards.
Link to originalSolution
05
08
Normal approximation - Heads v. tails
Flip a coin 10,000 times. Let
measure the number of heads, and measure the number of tails. Estimate the probability that and are within 100 of each other. Hint: Write an inequality for the condition, then sub a formula for
in terms of . Link to originalSolution
07
Let
and be the number of heads and tails respectively. Then we have the following two conditions: Thus,
. Let be the event that the -th flip is a head. Then for each and . Thus, by CLT,
. By the normal approximation, using the continuity correction, we have:
Link to original
06
05
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? Link to originalSolution
08
(a)
Let
be the indicator that a man was chosen in the -th period. Then for each , and the are independent for each . Let be the total number of times a man was chosen. We can use a similar argument to Problems 2 or 3, or we can simply use linearity of expectation: (b)
(1) Using the standard formula for the variance of a sum of random variables, we have:
Since the variables are identically distributed, their variances are equal. Thus,
(2) Now, the sum has
terms, and since each is identically distributed, each term is identical. Thus, for some fixed
.
(3) We then have
Finally,
Link to original, and thus Plugging these values in, we have .