5.2.4 Practice set: discrete distributions

Random variables, PMFs and CDFs

Random variables, PMFs and CDFs (lesson 5.2.1)

0 of 7 correct

1.
Toss a coin three times. $X$ is the number of tails minus the number of heads. Find $P(X = 1)$
2.
Roll two dice and let $N$ be the smaller number. Find $P(N = 1)$
3.
$p(k) = \frac{c}{k}$ for $k = 1, 2, 3$ is a pmf. Find $c$
4.
$X$ takes the values $-1$ , 0 and 2 with probabilities 0.3, 0.5 and 0.2. Find $F_X(1)$
5.
For the smaller of two dice, $P(N > k) = \left(\frac{6 - k}{6}\right)^2$ . Find $P(1 < N \le 3)$
6.
$X$ takes the values 1, 2, 4 and 5, with $F(1) = 0.2$ , $F(2) = 0.5$ , $F(4) = 0.9$ and $F(5) = 1$ . Find $P(2 \le X \le 4)$
7.
For the smaller of two dice, find $P(N \ge 2)$

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.

Bernoulli, binomial and geometric distributions

Bernoulli, binomial and geometric distributions (lesson 5.2.2)

0 of 7 correct

1.
$X \sim \operatorname{Bin}(5, 0.2)$ . Find $P(X = 1)$
2.
$X \sim \operatorname{Bin}(6, 0.5)$ . Find $P(X = 3)$
3.
Eight parts are tested; each is faulty with probability 0.1, independently. Find the probability that none is faulty (type powers with ^)
4.
$X \sim \operatorname{Bin}(4, 0.3)$ . Find $P(X \ge 3)$
5.
$X \sim \operatorname{Geom}(0.25)$ counts the trials up to the first success. Find $P(X = 3)$
6.
$X \sim \operatorname{Geom}(0.1)$ . Find $P(X > 10)$ (type powers with ^)
7.
$Y$ counts the failures before the first success, with $p = 0.5$ . Find $P(Y = 2)$

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 Poisson distribution

The Poisson distribution (lesson 5.2.3)

0 of 7 correct

1.
$X \sim \operatorname{Poisson}(1.5)$ . Find $P(X = 2)$ (type e for $e$ )
2.
$X \sim \operatorname{Poisson}(4)$ . Find $P(X \le 1)$ exactly
3.
$X \sim \operatorname{Poisson}(6)$ . Find $\frac{P(X = 4)}{P(X = 3)}$
4.
A Poisson count with $\lambda = 3$ has two most likely values. What is the larger one?
5.
A Poisson count has $P(X = 0) = 0.2$ . Find $\lambda$ exactly
6.
Use the Poisson approximation for $P(X = 1)$ when $X \sim \operatorname{Bin}(1000, 0.003)$ (type e for $e$ )
7.
Typos occur at 2 per page on average. Find the probability of no typo in a 3-page letter (type e for $e$ )

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

Binomial and Poisson probabilities

Write three functions with the math module only. The tests compare them with scipy.stats.

binomial_pmf(k, n, p) returns $\binom{n}{k}p^k(1 - p)^{n - k}$ (lesson 5.2.2).
poisson_pmf(k, lam) returns $e^{-\lambda}\frac{\lambda^k}{k!}$ (lesson 5.2.3).
cdf(pmf, x) returns $P(X \le x)$ for a random variable with whole-number values from 0 up, whose pmf is the function pmf (lesson 5.2.1). It adds pmf(k) for $k = 0, 1, \ldots$ up to $x$ , and works for any number $x$ .

The rules, as in the lessons:

A pmf is 0 at any $k$ the variable cannot take. That means a negative $k$ , a $k$ that is not whole, such as 2.5, and for the binomial a $k$ above $n$ .
Raise ValueError if $p$ is outside $[0, 1]$ , if $n$ is not a whole number from 0 up, or if $\lambda \le 0$ . Both $p = 0$ and $p = 1$ are allowed.
Some tests use large numbers, such as $\binom{2000}{1000}$ and $150^{200}$ , which are too big for a float. Compute the logarithm of the pmf first and take math.exp at the end; math.lgamma(k + 1) is $\ln k!$ .

See the tests

import math


def test_binomial_value():
    """Bin(10, 0.3) gives P(X = 3) = 0.2668"""
    expect_close(binomial_pmf(3, 10, 0.3), 0.2668279319999998, what="binomial_pmf(3, 10, 0.3)", rel=1e-9)


def test_binomial_sums_to_one():
    """The pmf of Bin(12, 0.35) adds up to 1 over k = 0 to 12"""
    expect_close(sum(binomial_pmf(k, 12, 0.35) for k in range(13)), 1.0, what="sum of the pmf", rel=1e-9)


def test_binomial_outside():
    """Bin(5, 0.5) is 0 at k = −1, 6 and 2.5"""
    for k in (-1, 6, 2.5):
        expect_close(binomial_pmf(k, 5, 0.5), 0.0, what=f"binomial_pmf({k}, 5, 0.5)")


def test_binomial_certain():
    """With p = 0 there are no successes; with p = 1 every trial succeeds"""
    expect_close(binomial_pmf(0, 5, 0), 1.0, what="binomial_pmf(0, 5, 0)")
    expect_close(binomial_pmf(2, 5, 0), 0.0, what="binomial_pmf(2, 5, 0)")
    expect_close(binomial_pmf(5, 5, 1), 1.0, what="binomial_pmf(5, 5, 1)")
    expect_close(binomial_pmf(2, 5, 1), 0.0, what="binomial_pmf(2, 5, 1)")


def test_binomial_large():
    """Bin(2000, 0.5) gives P(X = 1000) = 0.01784, though C(2000, 1000) is too big for a float"""
    expect_close(binomial_pmf(1000, 2000, 0.5), 0.017839011145854313, what="binomial_pmf(1000, 2000, 0.5)", rel=1e-9)


def test_binomial_bad_inputs():
    """p = 1.5, p = −0.1 and n = 2.5 raise ValueError"""
    for args in ((2, 5, 1.5), (2, 5, -0.1), (2, 2.5, 0.5)):
        try:
            binomial_pmf(*args)
        except ValueError:
            continue
        raise AssertionError(f"expected ValueError for binomial_pmf{args}")


def test_poisson_values():
    """Poisson(2) gives P(X = 2) = 2e⁻² and Poisson(3) gives P(X = 0) = e⁻³"""
    expect_close(poisson_pmf(2, 2), 2 * math.exp(-2), what="poisson_pmf(2, 2)", rel=1e-9)
    expect_close(poisson_pmf(0, 3), math.exp(-3), what="poisson_pmf(0, 3)", rel=1e-9)


def test_poisson_outside():
    """Poisson(2) is 0 at k = −1 and 1.5"""
    expect_close(poisson_pmf(-1, 2), 0.0, what="poisson_pmf(-1, 2)")
    expect_close(poisson_pmf(1.5, 2), 0.0, what="poisson_pmf(1.5, 2)")


def test_poisson_large():
    """Poisson(150) gives P(X = 200) = 1.5038e-05, though 150²⁰⁰ is too big for a float"""
    expect_close(poisson_pmf(200, 150), 1.5038027878284683e-05, what="poisson_pmf(200, 150)", rel=1e-8)


def test_poisson_bad_rate():
    """A rate of 0 or below raises ValueError"""
    for lam in (0, -1):
        try:
            poisson_pmf(1, lam)
        except ValueError:
            continue
        raise AssertionError(f"expected ValueError for poisson_pmf(1, {lam})")


def test_cdf_binomial():
    """For Bin(10, 0.3), F(2) = F(2.5) = 0.3828, F(−0.5) = 0 and F(15) = 1"""
    pmf = lambda k: binomial_pmf(k, 10, 0.3)
    expect_close(cdf(pmf, 2), 0.3827827863999998, what="cdf at 2", rel=1e-9)
    expect_close(cdf(pmf, 2.5), 0.3827827863999998, what="cdf at 2.5", rel=1e-9)
    expect_close(cdf(pmf, -0.5), 0.0, what="cdf at -0.5")
    expect_close(cdf(pmf, 15), 1.0, what="cdf at 15", rel=1e-9)


def test_cdf_interval():
    """For Poisson(2), P(2 < X ≤ 5) = F(5) − F(2) = 0.3068"""
    pmf = lambda k: poisson_pmf(k, 2)
    expect_close(cdf(pmf, 5) - cdf(pmf, 2), 0.30675997533632215, what="F(5) - F(2)", rel=1e-9)


def test_cdf_geometric():
    """Rolls until a six (values from 1): F(6) = 1 − (5/6)⁶"""
    pmf = lambda k: (5 / 6) ** (k - 1) / 6 if k >= 1 else 0.0
    expect_close(cdf(pmf, 6), 1 - (5 / 6) ** 6, what="cdf at 6", rel=1e-9)

Random variables, PMFs and CDFs

Bernoulli, binomial and geometric distributions

The Poisson distribution

Coding exercise

Free further reading