5.4.5 Practice set: continuous distributions

Densities and the uniform distribution

Densities, distribution functions and the uniform distribution (lesson 5.4.1)

0 of 6 correct

1.
Find the constant $c$ that makes $f(x) = c(4 - x)$ on $[0, 4]$ a density
2.
$X$ has density $3x^2$ on $[0, 1]$ . Find $P(X > 0.5)$
3.
$X$ has distribution function $F(x) = 1 - \frac{1}{x^2}$ for $x \ge 1$ . Find $P(2 < X \le 4)$
4.
$X$ is uniform on $[-1, 3]$ . Find $\operatorname{Var}(X)$
5.
$X$ is uniform on $[0, 3]$ . Find $\mathbb{E}[X^2]$
6.
A density equals 2 on $[0, 0.5]$ . Find $P(0.2 < X < 0.3)$

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 exponential distribution

The exponential distribution (lesson 5.4.2)

0 of 6 correct

1.
A wait is exponential with rate 0.5 a year. Find $P(X > 4)$ (type e^(…))
2.
A wait is exponential with mean 8 minutes. Find the probability it is under 4 minutes
3.
Find the median of an exponential wait with mean 10 minutes (type ln)
4.
A wait has mean 8 minutes, exponential, and you have already waited 10. Find the probability of waiting at least 8 more
5.
Requests arrive at 6 an hour. Find the probability of none in the next 10 minutes
6.
rng.exponential(2, size=1000) draws exponential waits. What is their rate $\lambda$ ?

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 normal distribution

The normal distribution (lesson 5.4.3)

0 of 6 correct

1.
$X \sim \mathcal{N}(100, 15^2)$ . Find the z-score of 130
2.
$X \sim \mathcal{N}(100, 15^2)$ . Use the 68–95–99.7 rule to find $P(X > 130)$ (5 decimal places)
3.
$X \sim \mathcal{N}(100, 15^2)$ and $z_{0.95} \approx 1.645$ . Find the 0.95-quantile of $X$
4.
$X \sim \mathcal{N}(\mu, 2^2)$ and $P(X < 0) = 0.025$ . With $z_{0.975} \approx 1.96$ , find $\mu$
5.
$X \sim \mathcal{N}(10, 3^2)$ and $Y \sim \mathcal{N}(5, 4^2)$ are independent. Find $\operatorname{Var}(2X - Y)$
6.
$X \sim \mathcal{N}(0, 4)$ and $N(1) \approx 0.8413$ . Find $P(X > 2)$ (4 decimal places)

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.

Lognormal, Student t and chi-squared

The lognormal, Student t and chi-squared distributions (lesson 5.4.4)

0 of 6 correct

1.
$\ln X \sim \mathcal{N}(2, 0.5^2)$ . Find the median of $X$ (type e^(…))
2.
$\ln X \sim \mathcal{N}(2, 0.5^2)$ . Find the mean of $X$ (type e^(…))
3.
$\ln X \sim \mathcal{N}(0.1, 0.2^2)$ and $\ln Y \sim \mathcal{N}(0.05, 0.1^2)$ , independent. Find the median of $XY$ (type e^(…))
4.
Find the variance of a Student t with 5 degrees of freedom
5.
$V \sim \chi^2_8$ . Find $\operatorname{Var}(V)$
6.
$\ln X \sim \mathcal{N}(0, 1)$ and $N(0.5) \approx 0.6915$ . Find $P(X > \mathbb{E}[X])$ (4 decimal places)

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

A quantile function for any distribution

Write quantile(cdf, p, lo, hi, tol=1e-10). It returns the $p$ -quantile of a distribution whose distribution function is cdf: the $x$ in $[\text{lo}, \text{hi}]$ with $F(x) = p$ (lesson 5.4.1).

Find it by bisection (lesson 1.3.2) on $F(x) - p$ : halve the bracket until it is narrower than tol, then return its midpoint.
Raise ValueError unless $0 < p < 1$ : the 0- and 1-quantiles can be infinite.
Raise ValueError if the bracket does not contain the quantile, that is if $F(\text{lo}) > p$ or $F(\text{hi}) < p$ .

The tests compare your answers with SciPy's ppf for a uniform, an exponential, a normal, a lognormal and a Student t. One bracket, $[-1, 1]$ for the 0.99-quantile of the standard normal, is too narrow.

See the tests

import math

from scipy.stats import expon, lognorm, norm, t, uniform


def test_uniform():
    """The 0.25-quantile of the uniform distribution on [2, 6] is 3"""
    expect_close(quantile(uniform(loc=2, scale=4).cdf, 0.25, 2, 6), 3.0, what="quantile of U(2, 6) at 0.25", abs_tol=1e-8)


def test_exponential_median():
    """The median of an exponential with rate 0.5 is 2 ln 2, about 1.386294"""
    expect_close(quantile(expon(scale=2).cdf, 0.5, 0, 50), 2 * math.log(2), what="median of Exp(0.5)", abs_tol=1e-8)


def test_normal_low_quantile():
    """The 0.01-quantile of N(500, 4²) is about 490.6946, as norm.ppf says"""
    X = norm(500, 4)
    expect_close(quantile(X.cdf, 0.01, 450, 550), X.ppf(0.01), what="0.01-quantile of N(500, 16)", abs_tol=1e-7)


def test_lognormal():
    """The 0.95-quantile of a lognormal with σ = 0.5 and median 1 is about 2.276017"""
    X = lognorm(s=0.5)
    expect_close(quantile(X.cdf, 0.95, 0, 20), X.ppf(0.95), what="0.95-quantile of the lognormal", abs_tol=1e-7)


def test_student_t():
    """The 0.975-quantile of a Student t with 4 degrees of freedom is about 2.776445"""
    expect_close(quantile(t(4).cdf, 0.975, -50, 50), t(4).ppf(0.975), what="0.975-quantile of t(4)", abs_tol=1e-7)


def test_tolerance():
    """A looser tolerance still lands within it: the standard normal's 0.975-quantile to 0.001"""
    x = quantile(norm.cdf, 0.975, -10, 10, tol=1e-3)
    if abs(x - norm.ppf(0.975)) > 1e-3:
        raise AssertionError(f"expected within 0.001 of 1.959964, got {x}")


def test_p_outside():
    """p of 0, 1 or 1.2 raises ValueError: those quantiles can be infinite or do not exist"""
    for p in [0, 1, 1.2]:
        try:
            quantile(norm.cdf, p, -10, 10)
        except ValueError:
            continue
        raise AssertionError(f"expected ValueError for p = {p}")


def test_bracket_too_narrow():
    """The 0.99-quantile of the standard normal, 2.326, is not in [−1, 1]: ValueError"""
    try:
        quantile(norm.cdf, 0.99, -1, 1)
    except ValueError:
        return
    raise AssertionError("expected ValueError: F(1) = 0.8413 is below 0.99, so [-1, 1] misses the quantile")

Densities and the uniform distribution

The exponential distribution

The normal distribution

Lognormal, Student t and chi-squared

Coding exercise

Free further reading