Lesson test: The laws of large numbers

1.[2p]
Which earlier result supplies the whole proof of the weak law?

Chebyshev's inequality, applied to an average whose variance is $σ^{2} / n$

Correct

The central limit theorem

The answer is: Chebyshev's inequality, applied to an average whose variance is $\sigma^2/n$

Bayes' theorem

The answer is: Chebyshev's inequality, applied to an average whose variance is $\sigma^2/n$

The union bound

The answer is: Chebyshev's inequality, applied to an average whose variance is $\sigma^2/n$

2.[3p]
Measurements have standard deviation $1$ unit. What does Chebyshev give as an upper bound on $P (| {\overline{X}}_{n} - μ | \geq 0.1)$ after $1000$ repeats?

CorrectNot quite: 0.1

3.[2p]
After ten tails in a row, a fair coin is more likely than not to land heads on the next toss.

True

The answer is: False

False

Correct

4.[3p]
How does a run of ten tails stop mattering as tossing continues?

The deficit of ten stays, and is divided by an ever larger number of tosses

Correct

Later tosses favour heads until the deficit is cancelled

The answer is: The deficit of ten stays, and is divided by an ever larger number of tosses

The coin's probability of heads rises slightly

The answer is: The deficit of ten stays, and is divided by an ever larger number of tosses

It does not: the fraction never returns towards a half

The answer is: The deficit of ten stays, and is divided by an ever larger number of tosses

5.[3p]
What does the central limit theorem require of the individual variables being summed?

Independence, a common distribution, and a finite variance

Correct

That they be normally distributed

The answer is: Independence, a common distribution, and a finite variance

That they be symmetric about their mean

The answer is: Independence, a common distribution, and a finite variance

That they be discrete

The answer is: Independence, a common distribution, and a finite variance

6.[3p]
Ten fair dice are rolled and totalled. What is the standard deviation of the total? Give the answer to four decimal places.

CorrectNot quite: 5.4006

7.[3p]
A poll of $1000$ people is taken. What is the worst-case $95$ percent margin of error, as a decimal fraction? Give the answer to three decimal places.

CorrectNot quite: 0.031

8.[3p]
How many people must be polled for a worst-case $95$ percent margin of error of one percentage point?

CorrectNot quite: 9604

9.[3p]
What happens when you average $n$ independent Cauchy variables?

Nothing: the average has the same distribution as a single one, however large $n$ is

Correct

The average converges to the mean, but more slowly than usual

The answer is: Nothing: the average has the same distribution as a single one, however large $n$ is

The average becomes normal, as the central limit theorem requires

The answer is: Nothing: the average has the same distribution as a single one, however large $n$ is

The average converges only if $n$ is even

The answer is: Nothing: the average has the same distribution as a single one, however large $n$ is

1.[2p] Which earlier result supplies the whole proof of the weak law?

2.[3p] Measurements have standard deviation 1 unit. What does Chebyshev give as an upper bound on P(|X‾n-μ|≥0.1) after 1000 repeats?

3.[2p] After ten tails in a row, a fair coin is more likely than not to land heads on the next toss.

4.[3p] How does a run of ten tails stop mattering as tossing continues?

5.[3p] What does the central limit theorem require of the individual variables being summed?

6.[3p] Ten fair dice are rolled and totalled. What is the standard deviation of the total? Give the answer to four decimal places.

7.[3p] A poll of 1000 people is taken. What is the worst-case 95 percent margin of error, as a decimal fraction? Give the answer to three decimal places.

8.[3p] How many people must be polled for a worst-case 95 percent margin of error of one percentage point?

9.[3p] What happens when you average n independent Cauchy variables?

1.[2p]
Which earlier result supplies the whole proof of the weak law?

2.[3p]
Measurements have standard deviation $1$ unit. What does Chebyshev give as an upper bound on $P (| {\overline{X}}_{n} - μ | \geq 0.1)$ after $1000$ repeats?

3.[2p]
After ten tails in a row, a fair coin is more likely than not to land heads on the next toss.

4.[3p]
How does a run of ten tails stop mattering as tossing continues?

5.[3p]
What does the central limit theorem require of the individual variables being summed?

6.[3p]
Ten fair dice are rolled and totalled. What is the standard deviation of the total? Give the answer to four decimal places.

7.[3p]
A poll of $1000$ people is taken. What is the worst-case $95$ percent margin of error, as a decimal fraction? Give the answer to three decimal places.

8.[3p]
How many people must be polled for a worst-case $95$ percent margin of error of one percentage point?

9.[3p]
What happens when you average $n$ independent Cauchy variables?