5.5.4 Practice set: joint distributions

Joint, marginal and conditional distributions

Several questions use this joint pmf of $X$ and $Y$ :

	$y = 0$	$y = 1$	$y = 2$
$x = 0$	0.20	0.10	0.10
$x = 1$	0.05	0.25	0.30

Joint, marginal and conditional distributions (lesson 5.5.1)

0 of 9 correct

1.
In the table, find $P(X = 1, Y \ge 1)$
2.
In the table, find the marginal probability $p_Y(2)$
3.
In the table, find $P(Y = 2 \mid X = 1)$
4.
In the table, find the conditional mean $\mathbb{E}[Y \mid X = 0]$
5.
In the table, find $P(X = 1 \mid Y = 0)$
6.
$f(x, y) = \frac{3}{2}(x^2 + y^2)$ on the unit square. Find the marginal density $f_X(0.5)$
7.
For the same density, find the conditional density $f_{Y \mid X}(1 \mid 0)$
8.
$f(x, y) = 8xy$ on the triangle $0 < y < x < 1$ . Find the marginal density $f_X(0.5)$
9.
Is the table above the joint pmf of independent variables? Type 1 for yes, 0 for no

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.

Covariance and correlation

Covariance, correlation and independence (lesson 5.5.2)

0 of 9 correct

1.
In the table above, find $\mathbb{E}[XY]$
2.
In the table above, $\mathbb{E}[X] = 0.6$ and $\mathbb{E}[Y] = 1.15$ . Find $\operatorname{Cov}(X, Y)$
3.
$\sigma_X = 0.2$ , $\sigma_Y = 0.3$ and $\rho = -0.5$ . Find $\operatorname{Var}(X + Y)$
4.
$\operatorname{Var}(X) = 1$ , $\operatorname{Var}(Y) = 2$ and $\operatorname{Cov}(X, Y) = 0.5$ . Find $\operatorname{Var}(X - 2Y)$
5.
$X$ and $Y$ have correlation 0.4. Find the correlation of $1 - 2X$ and $3Y$
6.
Find the sample correlation of the pairs $(1, 1)$ , $(2, 3)$ , $(3, 2)$ and $(4, 4)$
7.
np.var([1, 2, 3, 4]) returns 1.25. What does np.var([1, 2, 3, 4], ddof=1) return?
8.
Two random variables have standard deviations 0.1 and 0.2 and correlation 0.5. Find the entry $\Sigma_{12}$ of their covariance matrix
9.
$X$ is $-1$ , 0 or 1, each with probability $\frac{1}{3}$ , and $Y = X^2$ . 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.

The multivariate normal

The multivariate normal distribution (lesson 5.5.3)

0 of 8 correct

1.
$\mathbf{X} = L\mathbf{Z}$ with $L = \begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}$ . Find $\operatorname{Var}(X_2)$
2.
For the same $L$ , find the correlation of $X_1$ and $X_2$ (type sqrt)
3.
For the same $L$ , find the density of $\mathbf{X}$ at its mean (type pi for $\pi$ )
4.
$\mathbf{X} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$ with $\Sigma = \begin{pmatrix} 1 & 0.5 \\ 0.5 & 2 \end{pmatrix}$ . Find $\operatorname{Var}(2X_1 + X_2)$
5.
With the same $\Sigma$ and $\boldsymbol{\mu} = (1, 2)$ , $X_1 - X_2$ is normal with mean $-1$ . How many of its standard deviations is 0 above its mean? (type sqrt)
6.
A bivariate normal has $\Sigma$ with eigenvalues 1 and 4. Find the long semi-axis of the ellipse $Q = 9$
7.
A bivariate normal: find $P(Q \le 2)$ . Type an exact answer
8.
$Z$ is standard normal and $S$ an independent fair sign, $\pm 1$ . Find $P(Z + SZ > 0)$

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

Moments of a joint pmf, and a test of independence

A joint pmf is stored as a 2-D NumPy array p, with p[i, j] $= P(X = x_i, Y = y_j)$ for the values xs[i] and ys[j]. Write two functions, as in lessons 5.5.1 and 5.5.2.

joint_moments(p, xs, ys) returns the tuple (mean_x, mean_y, var_x, var_y, cov, rho), from the marginals and $\mathbb{E}[XY]$ .
is_independent(p, tol=1e-12) returns True when every cell is within tol of the product of its two marginals, and False otherwise.

Both raise ValueError if p is not 2-D, has a negative entry, or does not add up to 1 within $10^{-9}$ . joint_moments also raises it if the sizes of xs and ys do not match p. It raises it too if either variable is constant, since then $\rho$ is undefined.

The tests use lesson 5.5.1's three coins, the table above and case study 3.1.4's four values with covariance 0. They also try an independent table and each bad input.

See the tests

import numpy as np

COINS = np.array([[1, 1, 0], [1, 2, 1], [0, 1, 1]]) / 8
TABLE = np.array([[0.20, 0.10, 0.10], [0.05, 0.25, 0.30]])
FOUR = np.array([[0, 0.25], [0.25, 0], [0.25, 0], [0, 0.25]])  # X = 1..4, Y = 2 or 4


def _raises(f, *args, what=""):
    try:
        f(*args)
    except ValueError:
        return
    raise AssertionError(f"expected ValueError: {what}")


def _moments(*args):
    got = joint_moments(*args)
    assert got is not None, "joint_moments returned None (did you forget to return?)"
    return got


def test_coins():
    """Lesson 5.5.1's three coins: means 1, variances 1/2, covariance 1/4, correlation 1/2"""
    expect_close(joint_moments(COINS, [0, 1, 2], [0, 1, 2]), (1, 1, 0.5, 0.5, 0.25, 0.5), what="joint_moments(coin table)")


def test_table():
    """The practice table: covariance 0.16 and correlation about 0.4123"""
    got = joint_moments(TABLE, [0, 1], [0, 1, 2])
    expect_close(got, (0.6, 1.15, 0.24, 0.6275, 0.16, 0.41229445762405846), what="joint_moments(practice table)")


def test_other_values():
    """With the values 10, 20 and 30 for Y: mean 21.5, variance 62.75, covariance 1.6, the same correlation"""
    got = _moments(TABLE, [0, 1], [10, 20, 30])
    expect_close(got, (0.6, 21.5, 0.24, 62.75, 1.6, 0.41229445762405846), what="joint_moments with ys = (10, 20, 30)")


def test_zero_covariance():
    """Case study 3.1.4's four values: covariance 0, yet not independent"""
    got = _moments(FOUR, [1, 2, 3, 4], [2, 4])
    expect_close(got[4], 0.0, what="covariance of the four values", abs_tol=1e-12)
    expect_equal(is_independent(FOUR), False, what="is_independent(four values)")


def test_dependent():
    """The coin table and the practice table are not independent"""
    expect_equal(is_independent(COINS), False, what="is_independent(coin table)")
    expect_equal(is_independent(TABLE), False, what="is_independent(practice table)")


def test_independent():
    """A table built as a product of two marginals is independent, and its covariance is 0"""
    p = np.outer([0.4, 0.6], [0.25, 0.5, 0.25])
    expect_equal(is_independent(p), True, what="is_independent(product table)")
    expect_close(_moments(p, [0, 1], [0, 1, 2])[4], 0.0, what="covariance of the product table", abs_tol=1e-12)


def test_tolerance():
    """A table off a product by 0.001 is independent only with a looser tolerance"""
    p = np.outer([0.5, 0.5], [0.5, 0.5]) + np.array([[0.001, -0.001], [-0.001, 0.001]])
    expect_equal(is_independent(p), False, what="is_independent with the default tolerance")
    expect_equal(is_independent(p, tol=0.01), True, what="is_independent with tol=0.01")


def test_bad_tables():
    """A negative entry, a total that is not 1, or a 1-D array raises ValueError"""
    _raises(joint_moments, [[0.6, -0.1], [0.3, 0.2]], [0, 1], [0, 1], what="a negative probability")
    _raises(joint_moments, [[0.3, 0.3], [0.3, 0.3]], [0, 1], [0, 1], what="probabilities adding up to 1.2")
    _raises(joint_moments, [0.5, 0.5], [0, 1], [0], what="a 1-D array")
    _raises(is_independent, [[0.3, 0.3], [0.3, 0.3]], what="is_independent with a total of 1.2")


def test_bad_sizes():
    """Values that do not match the table's shape raise ValueError"""
    _raises(joint_moments, TABLE, [0, 1, 2], [0, 1, 2], what="three values of x for two rows")


def test_constant():
    """A variable that takes one value has no correlation: ValueError"""
    _raises(joint_moments, [[0.3, 0.7]], [5], [0, 1], what="X is always 5")

Joint, marginal and conditional distributions

Covariance and correlation

The multivariate normal

Coding exercise

Free further reading