01
Deviation estimation - Exponential
Let
with . (a) Compute the Markov bound on
. (b) Compute the Chebyshev bound on
. (c) Find the exact value of
and compare with yours answers in (a) and (b).
Solution
01
(a)
. By Markov’s Inequality,
.
(b)
. Chebyshev Inequality: where such that . Thus,
.
(c)
Since
is exponential: Clearly, the actual answer is quite small compared to our previously found upper bounds.
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02
Deviation estimation - How many samples required?
Suppose the expected value of a score on the Probability final exam is 80 and the variance is 10. Assume the students’ scores are independent.
How many students must take the exam before the average score in the class is known to lie within 5 points of 80 with a probability of 90%? What about 95%?
03
Deviation estimation - Factory production
Suppose a factory produces an average of
items per week. (a) How likely is it that more than 75 items are produced this week? (Find an upper bound.)
(b) Suppose the variance is known to be 25. Now what can you say about (a)? (Hint: Monotonicity.)
(c) What do you know about the probability that the number of items produced differs from the average by at most 10?
Solution
03
(a) Let
be the number of items produced in a week. We know . By Markov’s Inequality,
(b) By the Chebyshev Inequality,
(c) Similarly, by the Chebyshev Inequality:
Our inequality switches to
Link to originalsince the Chebyshev Inequality gives an upper bound for , and we have a negative sign in front of the concerned quantity.
04
Random walk forward
You play a game where you roll a die, and if the outcome is 1 or 2 you take a step forward, otherwise you take two steps forward. Let
be your position (measured in steps forward) after playing the game times. (a) Estimate
using a normal approximation for a certain relevant binomial distribution. (b) Find
and . Hint: Rewrite the conditions into a form where you can apply the Law of Large Numbers.
05
Train arrivals
The time between train arrivals at a station is exponentially distributed with a mean of 5 minutes. Therefore, the arrival time of the
train, , can be represented as where each is independent of the others and is exponentially distributed with a mean of 5 minutes. Suppose there are enough customers waiting to fill 3 trains. (a) Find the mean and variance of
, the time elapsed when the third train arrives. (b) Use Chebyshev’s inequality to find an upper bound for the probability that more than 35 minutes will pass before all customers can board a train.
Solution
08
(a)
(b)
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06
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.
(a) Use Markov’s Inequality to find an upper bound for the probability that the average age of the students in the random sample is more than 23 years.
(b) What is the probability that the total age of the students in the random sample is less than 1472 years?
Solution
10
(a)
(b)
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07
Math contest scores
At a high school math competition, students take a test with 10 questions. Each question is worth one point and the probability of a student getting any one question correct is 0.55, independent of the other questions.
(a) Find the variance of
, the average score for 15 students. (b) Use the Law of Large Numbers to find an upper bound for the probability that
is greater than 6.
Solution
09
(a)
(b)
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08
Normal v. Chebyshev - Jewelry insurance
A jewelry insurance provider has 2500 customers. The expected payout to a customer each year is $1000 with a standard deviation of $900.
What premium should be charged to each customer to ensure that the premiums will cover the claims, with probability at least 99.9%?
(a) Solve the problem using a normal approximation.
(b) Solve the problem using Chebyshev’s inequality.
09
Chebyshev
Suppose
is an RV with and . Use Chebyshev’s inequality to find:
(a) A lower bound for
. (b) An upper bound for
.