5.3.4 Practice set: expectation and moments

Expectation and LOTUS

Expectation and LOTUS (lesson 5.3.1)

0 of 7 correct

1.
$X$ is 1, 2 or 5 with probabilities 0.2, 0.5 and 0.3. Find $\mathbb{E}[X]$
2.
A raffle ticket costs 2 dollars, and 1 ticket in 500 wins 600 dollars. Find the expected net gain per ticket, in dollars
3.
A shot scores with probability 0.25, independently each time. On average, how many shots does it take to score once?
4.
$X$ is 1, 2 or 5 with probabilities 0.2, 0.5 and 0.3. Find $\mathbb{E}[X^2]$
5.
$X$ is $-2$ , $-1$ , 0, 1 or 2, each with probability $\frac{1}{5}$ . Find $\mathbb{E}\big[\lvert X \rvert\big]$ as a fraction
6.
$X$ is 0, 1 or 2 with probabilities $\frac{1}{4}$ , $\frac{1}{2}$ and $\frac{1}{4}$ , so $\mathbb{E}[X] = 1$ . Find $\mathbb{E}[2^X]$ as a fraction
7.
A die game pays 12 dollars on a six and nothing otherwise. Find the expected payout, in dollars

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.

Variance and linearity

Variance and linearity (lesson 5.3.2)

0 of 7 correct

1.
$X$ is 1, 2 or 5 with probabilities 0.2, 0.5 and 0.3, so $\mathbb{E}[X] = 2.7$ and $\mathbb{E}[X^2] = 9.7$ . Find $\operatorname{Var}(X)$
2.
A temperature in degrees Celsius has standard deviation 5. In Fahrenheit it is $1.8C + 32$ . Find its standard deviation in Fahrenheit
3.
A die is rolled 30 times. Find the variance of the number of sixes, as a fraction
4.
Roll 4 dice and score their total minus 10. Find the expected score
5.
In a group of 30 people, each pair shares a birthday with probability $\frac{1}{365}$ . Find the expected number of pairs that share one, as a fraction
6.
$\operatorname{Var}(X) = 2.41$ . Find $\operatorname{Var}(4 - 2X)$
7.
$X$ and $Y$ are independent, with variances 3 and 5. Find $\operatorname{Var}(X - Y)$

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.

Skewness and kurtosis

Skewness and kurtosis (lesson 5.3.3)

0 of 7 correct

1.
$X$ has third central moment $\mathbb{E}[(X - \mu)^3] = 16$ and standard deviation 2. Find its skewness
2.
Find the skewness of a Bernoulli variable with $p = 0.25$ , exactly
3.
Find the kurtosis of a Bernoulli variable with $p = 0.25$ , as a fraction
4.
$X$ is 0 with probability 0.8, and $-\sqrt{5}$ or $\sqrt{5}$ with probability 0.1 each. Its mean is 0 and its variance 1. Find its kurtosis
5.
Find the excess kurtosis of that same $X$
6.
A data set has skewness $-0.31$ . Every value is multiplied by $-1$ . Find the new skewness
7.
scipy.stats.kurtosis(x, fisher=False) returns 6.5. What does scipy.stats.kurtosis(x) return?

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

The four moments of a discrete distribution

Write moments(values, probs). It takes the values of a discrete random variable and their probabilities, as lists or NumPy arrays. It returns a tuple (mean, variance, skewness, excess_kurtosis), as in lessons 5.3.1 to 5.3.3.

Use LOTUS for every moment: the mean, the variance $\mathbb{E}[(X - \mu)^2]$ , and $\mathbb{E}[Z^3]$ and $\mathbb{E}[Z^4] - 3$ with $Z = \frac{X - \mu}{\sigma}$ .
Raise ValueError if the two inputs have different lengths or are empty, or if any probability is negative. Also raise it if the probabilities do not add up to 1, allowing $10^{-9}$ for rounding.
Raise ValueError if the variance is 0, because then $Z$ is $\frac{0}{0}$ .

The tests use a die, a Bernoulli variable, lesson 5.3.3's family with kurtosis 10, a shifted and scaled die, and inputs that must be refused.

See the tests

import math


def _raises(values, probs):
    try:
        moments(values, probs)
    except ValueError:
        return True
    return False


def test_die():
    """A fair die has mean 3.5, variance 35/12, skewness 0 and excess kurtosis about -1.2686"""
    expect_close(moments([1, 2, 3, 4, 5, 6], [1 / 6] * 6), (3.5, 35 / 12, 0.0, -1.2685714285714285), what="moments(die)", rel=1e-9, abs_tol=1e-12)


def test_bernoulli():
    """A Bernoulli variable with p = 0.2 has skewness 1.5 and excess kurtosis 0.25"""
    expect_close(moments([0, 1], [0.8, 0.2]), (0.2, 0.16, 1.5, 0.25), what="moments([0, 1], [0.8, 0.2])", rel=1e-9, abs_tol=1e-12)


def test_heavy_tails():
    """0 with chance 0.9 and ±√10 with chance 0.05 each: variance 1, excess kurtosis 7"""
    a = math.sqrt(10)
    expect_close(moments([-a, 0, a], [0.05, 0.9, 0.05]), (0.0, 1.0, 0.0, 7.0), what="moments of lesson 5.3.3's family with q = 0.1", rel=1e-9, abs_tol=1e-12)


def test_shift_and_scale():
    """10X + 3 for a die: mean 38, variance 100 times 35/12, the same skewness and kurtosis"""
    expect_close(moments([13, 23, 33, 43, 53, 63], [1 / 6] * 6), (38.0, 3500 / 12, 0.0, -1.2685714285714285), what="moments(10X + 3)", rel=1e-9, abs_tol=1e-9)


def test_mirror():
    """-X flips the sign of the skewness and keeps the kurtosis: 0 or -1 with chances 0.8 and 0.2"""
    expect_close(moments([0, -1], [0.8, 0.2]), (-0.2, 0.16, -1.5, 0.25), what="moments([0, -1], [0.8, 0.2])", rel=1e-9, abs_tol=1e-12)


def test_bad_probabilities():
    """Probabilities that add up to 0.9, or include a negative one, raise ValueError"""
    assert _raises([1, 2, 3], [0.3, 0.3, 0.3]), "probabilities adding up to 0.9 should raise ValueError"
    assert _raises([1, 2, 3], [1.2, -0.1, -0.1]), "a negative probability should raise ValueError"


def test_bad_shapes():
    """Lists of different lengths, or empty lists, raise ValueError"""
    assert _raises([1, 2, 3], [0.5, 0.5]), "three values and two probabilities should raise ValueError"
    assert _raises([], []), "empty inputs should raise ValueError"


def test_no_spread():
    """A constant has variance 0, so its skewness is undefined: ValueError"""
    assert _raises([4, 4], [0.5, 0.5]), "a variable that is always 4 should raise ValueError"

Expectation and LOTUS

Variance and linearity

Skewness and kurtosis

Coding exercise

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