5.7.4 Practice set: inequalities and limit theorems

The Markov, Chebyshev and Jensen inequalities

The Markov, Chebyshev and Jensen inequalities (lesson 5.7.1)

0 of 9 correct

1.
$X \ge 0$ has mean 4. Bound $P(X \ge 20)$ with Markov's inequality
2.
A fair die shows $X$ . What bound does Markov's inequality give for $P(X \ge 4)$ ?
3.
$X$ has mean 100 and standard deviation 10. Bound $P(|X - 100| \ge 25)$ with Chebyshev's inequality
4.
By Chebyshev's inequality, at least what fraction of any distribution lies within 4 standard deviations of its mean?
5.
$\operatorname{Var}(X) = 9$ . For which distance $\varepsilon$ does Chebyshev's bound on $P(|X - \mu| \ge \varepsilon)$ equal 0.01?
6.
$X$ is 1 or 3 with equal chances. Find $\mathbb{E}\left[\frac{1}{X}\right] - \frac{1}{\mathbb{E}[X]}$
7.
$Y \sim \mathcal{N}(0.1, 0.09)$ . Find $\mathbb{E}[e^Y]$ exactly (type e)
8.
With $u(w) = \ln w$ , wealth is 50 or 200 with equal chances. Find the risk premium: expected wealth minus the certainty equivalent
9.
$X$ is $-2$ or 2 with equal chances, so $\mathbb{E}[X] = 0$ . Find $P(X \ge 2)$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

The law of large numbers

The law of large numbers (lesson 5.7.2)

0 of 8 correct

1.
Draws have standard deviation 8. Find the standard error of the mean of 64 of them
2.
A fair coin is scored 1 for heads and 0 for tails. Find the variance of the mean of 10 tosses
3.
A standard error is 0.4. How many times as many draws bring it down to 0.1?
4.
Draws have $\sigma = 3$ . Bound $P(|\bar{X}_{900} - \mu| \ge 0.5)$ with Chebyshev's inequality
5.
An event has probability 0.1. Find the standard error of its share in 900 independent trials
6.
A fair coin is tossed 400 times. Find the standard deviation of the number of heads minus 200
7.
A fair coin has shown 520 heads in 1,000 tosses. How many heads do you expect in the next 1,000?
8.
Draws have $\sigma = 6$ . Find the standard error of the mean of 36 of them

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

The central limit theorem

The central limit theorem (lesson 5.7.3)

0 of 8 correct

1.
Find the standard deviation of the sum of 300 independent uniforms on $[0, 1]$
2.
Draws have $\mu = 50$ and $\sigma = 20$ . Given $N(2) \approx 0.9772$ , estimate $P(\bar{X}_{100} < 46)$
3.
Draws have $\sigma = 10$ . Find the 95% half-width $1.96\frac{\sigma}{\sqrt{n}}$ for the mean of 400 of them
4.
Draws have $\sigma = 1$ . How many make the 95% half-width $\frac{1.96}{\sqrt{n}}$ equal to 0.049?
5.
A fair coin is tossed 400 times. Given $N(1.95) \approx 0.9744$ , estimate $P(\text{heads} \ge 220)$ with the half-step
6.
An estimate is good to 2 decimal places. About how many times the draws give one more?
7.
You draw 1,000 values from an exponential distribution. What is the skewness of the distribution of a single draw?
8.
The middle 90% of a single t draw with $\nu = 1$ has width 12.63. What is the width for the mean of 100 such draws?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

How many draws?

Write draws_needed(sigma, eps, delta). It returns the pair (n_chebyshev, n_normal). Each is a number of draws $n$ that keeps $P(|\bar{X}_n - \mu| \ge \varepsilon)$ at or below $\delta$ .

Here $\sigma$ = sigma is the standard deviation of one draw, $\varepsilon$ = eps and $\delta$ = delta.

n_chebyshev is the smallest whole $n$ with $\frac{\sigma^2}{n\varepsilon^2} \le \delta$ (lesson 5.7.2): a guarantee for every distribution.
n_normal is the smallest whole $n$ with $z\frac{\sigma}{\sqrt{n}} \le \varepsilon$ , where $z$ is norm.ppf(1 - delta / 2) (lesson 5.7.3): the central limit theorem's answer.
Both are at least 1; with sigma = 0 one draw is enough. Use math.ceil to round up.
Raise ValueError if sigma is negative, eps is 0 or below, or delta is not strictly between 0 and 1.

For a die and eps = 0.1, delta = 0.05, the answers are 5,834 and 1,121, as in lessons 5.7.2 and 5.7.3.

See the tests

import math


def test_die():
    """A die (sigma² = 35/12), within 0.1 with probability 0.95: 5834 rolls by Chebyshev, 1121 by the normal"""
    got = draws_needed(math.sqrt(35 / 12), 0.1, 0.05)
    expect_equal(tuple(got), (5834, 1121), what="draws_needed(sqrt(35/12), 0.1, 0.05)")


def test_share():
    """A fair coin's share (sigma = 0.5) within 0.015 with probability 0.95: 22223 and 4269 tosses"""
    got = draws_needed(0.5, 0.015, 0.05)
    expect_equal(tuple(got), (22223, 4269), what="draws_needed(0.5, 0.015, 0.05)")


def test_other_delta():
    """sigma = 2, eps = 0.3, delta = 0.1: the normal uses its 0.95-quantile 1.6449, giving 445 and 121"""
    got = draws_needed(2, 0.3, 0.1)
    expect_equal(tuple(got), (445, 121), what="draws_needed(2, 0.3, 0.1)")


def test_hundred_times():
    """One more digit (eps divided by 10) needs about 100 times the draws"""
    n1 = draws_needed(1.5, 0.05, 0.05)[1]
    n2 = draws_needed(1.5, 0.005, 0.05)[1]
    expect_close(n2 / n1, 100, what="ratio of draws for eps = 0.005 and eps = 0.05", rel=0.01)


def test_chebyshev_is_cautious():
    """Chebyshev's guarantee always asks for at least as many draws as the normal"""
    for delta in [0.5, 0.1, 0.05, 0.01, 0.001]:
        cheb, normal = draws_needed(1.0, 0.01, delta)
        assert cheb >= normal, f"with delta = {delta}, Chebyshev gave {cheb} but the normal gave {normal}"


def test_no_spread():
    """With sigma = 0 every draw is mu, so one draw is enough: (1, 1)"""
    expect_equal(tuple(draws_needed(0, 0.1, 0.05)), (1, 1), what="draws_needed(0, 0.1, 0.05)")


def test_whole_numbers():
    """Both answers are whole numbers of draws"""
    for n in draws_needed(3.0, 0.2, 0.05):
        assert isinstance(n, int), f"expected whole numbers (int), got {n!r} of type {type(n).__name__}"


def test_bad_inputs():
    """A negative sigma, eps of 0 or below, or delta outside (0, 1) raises ValueError"""
    cases = [
        ("sigma = -1", lambda: draws_needed(-1, 0.1, 0.05)),
        ("eps = 0", lambda: draws_needed(1, 0, 0.05)),
        ("eps = -0.1", lambda: draws_needed(1, -0.1, 0.05)),
        ("delta = 0", lambda: draws_needed(1, 0.1, 0)),
        ("delta = 1", lambda: draws_needed(1, 0.1, 1)),
        ("delta = 1.5", lambda: draws_needed(1, 0.1, 1.5)),
    ]
    for name, call in cases:
        try:
            call()
        except ValueError:
            continue
        raise AssertionError(f"expected ValueError for {name}")

The Markov, Chebyshev and Jensen inequalities

The law of large numbers

The central limit theorem

Coding exercise

Free further reading