5  Discrete Random Variables

• Roughly, a random variable assigns a number measuring some quantity of interest to each outcome of a random phenomenon.
• Mathematically, a random variable (RV) $X$ is a function that takes an outcome in the sample space as input and returns a real number as output
• The random variable itself is typically denoted with a capital letter ($X$); possible values of that random variable are denoted with lower case letters ($x$).
  • Think of the capital letter $X$ as a label standing in for a formula like “the number of heads in 4 flips of a coin” and
  • $x$ as a dummy variable standing in for a particular value like 3.
• Discrete random variables take at most countably many possible values (e.g., $0,1,2,\dots$). They are often counting variables (e.g., the number of Heads in 10 coin flips).
• Continuous random variables can take any real value in some interval (e.g., $[0,1]$, $[0,\mathrm{\infty })$, $(-\mathrm{\infty },\mathrm{\infty })$.). That is, continuous random variables can take uncountably many different values. Continuous random variables are often measurement variables (e.g., height, weight, income).
• A function of a random variable is also a random variable: if $X$ is a RV then so is $g(X)$
• Sums and products, etc., of random variables defined on the same sample space are random variables. If$X$ and $Y$ are RVs defined on the same sample space then so are $X+Y$, $X-Y$, $XY$
• If the sample space outcomes are represented by rows in a spreadsheet, then random variables correspond to columns.
• Expressions like $X=x$ or $\{X=x\}$ represent events: for which outcomes is the value of the random variable $X$ equal to the value $x$. (Remember, if the sample space outcomes are represented by rows in a spreadsheet, then an event corresponds to a subset of rows (outcomes) that satisfies some criteria.)
• The (probability) distribution of a collection of random variables identifies the possible values that the random variables can take and their relative likelihoods.
• We will see many ways of describing a distribution, depending on how many random variables are involved and their types (discrete or continuous).

Table 5.1: Table representing the sum ($X$) and larger ($Y$) of two rolls of a four-sided die

Outcome (First roll, second roll)	X (sum)	Y (max)
(1, 1)	2	1
(1, 2)	3	2
(1, 3)	4	3
(1, 4)	5	4
(2, 1)	3	2
(2, 2)	4	2
(2, 3)	5	3
(2, 4)	6	4
(3, 1)	4	3
(3, 2)	5	3
(3, 3)	6	3
(3, 4)	7	4
(4, 1)	5	4
(4, 2)	6	4
(4, 3)	7	4
(4, 4)	8	4

Example 5.1 Roll a four-sided die twice, and record the result of each roll in sequence. Let $X$ be the sum of the two dice, and let $Y$ be the larger of the two rolls (or the common value if both rolls are the same). Table 5.1 represents the possible outcomes and the values of the random variables $X$ and $Y$. Assume the die is fair and the rolls are independent.

1. Identify the event $\{X=4\}$ and compute its probability. Then interpret the probability both as a long relative frequency and as a relative likelihood.
   
   
   
   
   
2. Construct a table and plot of $\text{P}(X=x)$ for each possible value $x$ of $X$.
   
   
   
   
   
3. Construct a table and plot of $\text{P}(Y=y)$ for each possible value $y$ of $Y$.
   
   
   
   
   

Table 5.2: The marginal distribution of $X$, the sum of two rolls of a fair four-sided die.

x	P(X=x)
2	0.0625
3	0.1250
4	0.1875
5	0.2500
6	0.1875
7	0.1250
8	0.0625

Table 5.3: The marginal distribution of $Y$, the larger of two rolls of a fair four-sided die.

y	P(Y=y)
1	0.0625
2	0.1875
3	0.3125
4	0.4375

Figure 5.1: The marginal distribution of $X$, the sum of two rolls of a fair four-sided die.

Figure 5.2: The marginal distribution of $Y$, the larger of two rolls of a fair four-sided die.

5.1 Simulating from a Distribution

Example 5.2 Continuing Example 5.1.

1. Describe how you could simulate a single value of $Y$.
   
   
   
   
   
2. Construct a spinner (like from a kids game) to represent the distribution of $Y$.
   
   
   
   
   
3. Describe another way to simulate a single value of $Y$.
   
   
   
   
   
4. Describe how you could use simulation to approximate the distribution of $Y$.
   
   
   
   
   

Figure 5.3: Spinner representing the marginal distribution of $X$, the sum of two rolls of a fair four-sided die.

Figure 5.4: Spinner representing the marginal distribution of $Y$, the larger of two rolls of a fair four-sided die.

• Don’t confuse a random variable with its distribution!
  • A random variable measures a numerical quantity which depends on the outcome of a random phenomenon
  • The distribution of a random variable specifies the long run pattern of variation of values of the random variable over many repetitions of the underlying random phenomenon.
• Any marginal distribution can be represented by a single spinner (like from a kids game).
• In principle, there are always two ways of simulating a value $x$ of a random variable $X$.
  1. Simulate from the probability space. Simulate an outcome $\omega$ from the underlying probability space and set $x=X(\omega )$.
  2. Simulate from the distribution. Construct a spinner corresponding to the distribution of $X$ and spin it once to generate $x$.
• The second method requires that the distribution of $X$ is known. However, as we will see in many examples, it is common to specify the distribution of a random variable directly without defining the underlying probability space.

5.2 Probability Mass Functions

• The probability distribution of a single discrete random variable $X$ is often displayed in a table containing the probability of the event $\{X=x\}$ for each possible value $x$.
• In some cases, a distribution has a “formulaic” shape. For a discrete random variable $X$, the probability mass function (pmf) $p_X$ expresses $\text{P}(X=x)$ as a function of $x$: $p_X(x)=\text{P}(X=x)$.
• Be sure to specify the possible values of the random variable!
• Certain common distributions have special names and properties.

Example 5.3 Continuing Example 5.1.

1. Verify that the pmf of $X$ is $$p_X(x)=\frac{4-|x-5|}{16},x=2,3,\dots ,8$$
   
   
   
   
   
2. Verify that the pmf of $Y$ is $$p_Y(y)=\frac{2y-1}{16},y=1,2,3,4$$
   
   
   
   
   

5.3 Matching Problem

The “matching problem” involves $n$ distincts objects labeled $1,\dots ,n$ which are placed in $n$ distinct boxes labeled $1,\dots ,n$, with exactly one object placed in each box. Suppose the objects are placed in the boxes uniformly at random, so that any possible arrangement is equally likely. Let $X$ be the number of matches, that is, the number of objects for which their label matches the label of the box in which they are placed.

Table 5.4: Table representing the sample space in the matching problem with $n=4$.

Spot 1	Spot 2	Spot 3	Spot 4	X (number of matches)
1	2	3	4
1	2	4	3
1	3	2	4
1	3	4	2
1	4	2	3
1	4	3	2
2	1	3	4
2	1	4	3
2	3	1	4
2	3	4	1
2	4	1	3
2	4	3	1
3	1	2	4
3	1	4	2
3	2	1	4
3	2	4	1
3	4	1	2
3	4	2	1
4	1	2	3
4	1	3	2
4	2	1	3
4	2	3	1
4	3	1	2
4	3	2	1

Example 5.4 Consider the matching problem with $n=4$. There are $4!=24$ possible outcomes, displayed in Table 5.4.

1. Complete Table 5.4 to identify the value of $X$ for each outcome.
   
   
   
   
   
2. Identify the event $\{X=0\}$ and compute its probability. Then interpret the probability both as a long relative frequency and as a relative likelihood.
   
   
   
   
   
3. Construct a table, plot, and spinner representing the distribution of $X$.
   
   
   
   
   

# One repetition of the number of matches, for a given n
simulate_number_matches = function(n) {
  # sample(1:n) puts the values 1:n in random order
  # sample(1:n) == 1:n checks if each object in the shuffled order matches
  #   returns a logical/binary 1=TRUE, 0 = FALSE vector
  # count the number of matches by summing the 1/0's
  
  sum(sample(1:n) == 1:n)
}

# Many repetitions, for n = 4
N_rep = 10000
number_matches = replicate(N_rep, simulate_number_matches(4))

# Summarize the simulated values
plot(table(number_matches) / N_rep,
     type = "h",
     xlab = "Number of matches",
     ylab = "Simulated relative frequency")

Figure 5.5: Simulated distribution of $X$ (number of matches) in the matching problem with $n=4$.

Table 5.5: Theoretical distribution of $X$ (number of matches) in the matching problem with $n=4$.

x	P(X=x)
0	0.3750
1	0.3333
2	0.2500
4	0.0417

Figure 5.6: Spinner representing the theoretical distribution of $X$, the number of matches in the matching problem when $n=4$.

Example 5.5 Now consider the matching problem with general $n$. Finding the distribution of $X$ by enumerating the $n!$ possible outcomes as we did for $n=4$ is not feasible for general $n$. But we can approximate the distribution of $X$ with simulation. Simulation shows that the distribution of the number of matches $X$ is approximately the same for all $n$ (unless $n$ is very small, i.e., $n\le 5$). In particular, for any $n$, the probability mass function of $X$ is approximately $$p_X(x)=\frac{e^{-1}}{x!},x=0,1,2,\dots$$

1. Use this pmf to approximate the probability of at least one match, and compare to the simulation results for general $n$.
   
   
   
   
   
2. Interpret the probability both as a long relative frequency and as a relative likelihood.
   
   
   
   
   
3. Construct a table, plot, and spinner corresponding to the above pmf. Compare to the simulation results for general $n$.
   
   
   
   
   

x = 0:7

p_x = exp(-1) / factorial(x)

data.frame(x, p_x) |>
  kbl(digits = 6) |>
  kable_styling(fixed_thead = TRUE)

Table 5.6: Approximate distribution of number of matches in the matching problem for general $n$

x	p_x
0	0.367879
1	0.367879
2	0.183940
3	0.061313
4	0.015328
5	0.003066
6	0.000511
7	0.000073

plot(x, p_x,
     type = "h",
     xlab = "Number of matches (x)",
     ylab = "Approximate probability p(x)")

Figure 5.7: Approximate distribution of number of matches in the matching problem for general $n$

Figure 5.8: Spinner representing the approximate distribution of number of matches in the matching problem for general $n$

n = 10

# Method 1: Simulate from the probability space
number_matches = replicate(N_rep, simulate_number_matches(n))

# Summarize the simulate values
plot(table(number_matches) / N_rep,
     type = "h",
     xlab = "Number of matches",
     ylab = "Simulated relative frequency")

Figure 5.9: Simulated distribution of $X$ (number of matches) in the matching problem with $n=10$: By simulating outcomes and measuring $X$.

# Method 2: Simulate from the (approximate) distribution
x_values = 0:n

x = sample(x_values,
           size = N_rep,
           prob = exp(-1) / factorial(x_values),
           replace = TRUE)

# Summarize the simulate values
plot(table(number_matches) / N_rep,
     type = "h",
     xlab = "Number of matches",
     ylab = "Simulated relative frequency")

Figure 5.10: Simulated distribution of $X$ (number of matches) in the matching problem with $n=10$: By simulating directly from the approximate distribution.