In this topic you bound probabilities using only a mean or a variance. No distribution is given. You put an upper limit on a tail probability. You pick a sample size so the average is close to the true mean with high probability. A problem belongs here when it gives only a mean and maybe a variance, names no distribution, and asks "at most", "at least", or "how many samples". You will practice: Markov's inequality, Chebyshev's inequality, concentration of the sample mean, and the weak law of large numbers.
An office receives a random number X of emails per day; X is nonnegative with E[X] = 8. Using Markov's inequality, give the best upper bound on P(X ≥ 40).
Answer:
X is nonnegative with known mean, so Markov's inequality applies. No variance is given, so Markov is the only available tool.
Exam scores have mean 70 and variance 16. Use Chebyshev's inequality to bound P(|X − 70| ≥ 10).
Answer:
Apply Chebyshev's inequality with deviation 10.
A sensor takes 9 independent readings of the same quantity; each reading has variance 36. What is the variance of the average of the 9 readings?
Answer:
Independence lets variances add across the sum. Dividing the sum by n scales its variance by 1 over n squared.
Daily visits to a website have mean 200 and standard deviation 10. Use Chebyshev to give a lower bound on P(170 < X < 230). Give the answer as a fraction.
Answer:
The interval is the mean plus or minus 30, which is 3 standard deviations. Bound the deviation event with Chebyshev, then take the complement.
A bakery sells a random number X of cakes per day. X is nonnegative with E[X] = 5. Use Markov's inequality to give an upper bound on P(X ≥ 20).
Answer:
Apply Markov's inequality directly.
A student's commute time X has mean 30 minutes and variance 9. Use Chebyshev's inequality to give an upper bound on P(|X − 30| ≥ 9).
Answer:
Apply Chebyshev's inequality with mean 30 and variance 9.
An engineer tests 25 batteries. The lifetimes are iid, each with standard deviation 10 hours. Find the standard deviation of the sample mean of the 25 lifetimes.
Answer:
The standard deviation of the sample mean is the single-observation standard deviation divided by the square root of n.
A machine fills bottles with mean 500 ml and standard deviation 4 ml. Use Chebyshev's inequality to give an upper bound on the probability that a bottle's fill differs from 500 ml by at least 10 ml.
Answer:
The variance is 16. Apply Chebyshev with deviation 10.
A waiter's daily tips are iid with mean 40 dollars and variance 100. Let X̄ be the average of 25 days of tips. Use Chebyshev's inequality to give an upper bound on P(|X̄ − 40| ≥ 4).
Answer:
First find the variance of the sample mean, then apply Chebyshev to it.
The number of cars X passing a toll booth in an hour is nonnegative with E[X] = 12. Use Markov's inequality to give a lower bound on P(X < 60).
Answer:
Markov bounds the complement event, then subtract from 1.
A fair coin has just landed heads 10 times in a row. All flips are independent. A gambler says tails are now due, so the next flips must balance out. Let X be the number of heads in the next 100 flips. Compute E[X].
Answer:
Independence means past flips do not affect future flips. The law of large numbers describes long-run proportions and is not a correcting force, so the gambler is wrong.
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