7 Marginal Distributions

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).
In the context of multiple random variables, the distribution of any one of the random variables is called a marginal distribution.

7.1 Discrete random variables

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 expresses \(\text{P}(X=x)\) as a function of \(x\).

Example 7.1

Roll a fair four-sided die twice. 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).

Construct a table and plot displaying the marginal distribution of \(X\).
Construct a table and plot displaying the marginal distribution of \(Y\).
Describe the distribution of \(Y\) in terms of long run relative frequency.
Describe the distribution of \(Y\) in terms of relative degree of likelihood.

7.2 Simulating from a marginal distribution

Any marginal distribution can be represented by a single spinner.
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.

Example 7.2

Continuing Example Example 7.1.

Construct a spinner to represent the marginal distribution of \(X\).
How could you use the spinner from the previous part to simulate a value of \(X\).
Construct a spinner to represent the marginal distribution of \(Y\).
How could you use the spinner from the previous part to simulate a value of \(Y\).
Donny Don’t says: “Great! I can simulate an \((X, Y)\) pair just by spinning the \(X\)-spinner to generate \(X\) and the \(Y\)-spinner to generate \(Y\).” Is Donny correct? If not, can you help him see why not?

7.3 Continuous random variables

Simulated values of a continuous random variable are usually plotted in a histogram which groups the observed values into “bins” and plots densities or frequencies for each bin.
In a histogram areas of bars represent relative frequencies; the axis which represents the height of the bars is called “density”.
The probability that a continuous random variable \(X\) equals any particular value is 0. That is, if \(X\) is continuous then \(\text{P}(X=x)=0\) for all \(x\).
Even though any specific value of a continuous random variable has probability 0, intervals still can have positive probability.
In practical applications involving continuous random variables, “equal to” really means “close to”, and “close to” probabilities correspond to intervals which can have positive probability.
The marginal distribution of a continuous random variable can be described by a probability density function, for which areas under the density curve determine probabilities.

Example 7.3

Let \(U\) be a random variable with a Uniform(0, 1) distribution. Suppose we want to approximate \(\text{P}(U = 0.2)\); that is, \(\text{P}(U = 0.2000000000000000000000\ldots)\), the probability that \(U\) is equal to 0.2 with infinite precision.

Use simulation to approximate \(\text{P}(U = 0.2)\).
What do you notice? Why does this make sense?
Use simulation to approximate \(\text{P}(|U - 0.2|<0.005) = \text{P}(0.195 < U < 0.205)\), the probability that \(U\) rounded to two decimal places is equal to 0.2.
Use simulation to approximate \(\text{P}(|U - 0.2|<0.0005) = \text{P}(0.1995 < U < 0.2005)\), the probability that \(U\) rounded to three decimal places is equal to 0.2.
Explain why \(\text{P}(U = 0.2) = 0.\) Also explain, why this is not a problem in practical applications.

7.4 Normal distributions

Normal distributions follow a particular bell-shaped curve which corresponds to a very specific pattern of variation

Example 7.4

Consider the Normal(30, 10) spinner. If we spin the spinner many times:

About what percent of values would be below 30? Above 30?
About what percent of values would be between 20 and 30? Between 30 and 40?
How would the shape of the distribution below 30 compare to that above 30?
About what percent of values would be between 20 and 40?
About what percent of values would be between 10 and 50?

7.5 Percentiles

A distribution is characterized by its percentiles.
Roughly, the value \(x\) is the \(p\)th percentile (a.k.a. quantile) of a distribution of a random variable \(X\) if \(p\) percent of values of the variable are less than or equal to \(x\): \(\text{P}(X\le x) = p\).
A spinner basically describes a distribution by specifying all the percentiles. For example,
- The 25th percentile goes 25% of the way around the axis (at “3 o’clock”)
- The 50th percentile goes 50% of the way around the axis (at “6 o’clock”)
- The 75th percentile goes 75% of the way around the axis (at “9 o’clock”)

Example 7.5

Recall that the spinner that represents the Normal(30, 10) distribution. According to this distribution:

What percent of values are less than 23.26?
What is the 25th percentile?
What is the 75th percentile?
A value of 40 corresponds to what percentile?

7.6 Transformations

Many random variables are derived as transformations of other random variables.
A function of a random variable is a random variable: if \(X\) is a random variable and \(g\) is a function then \(Y=g(X)\) is a random variable.
In general, the distribution of \(g(X)\) will have a different shape than the distribution of \(X\). The exception is when \(g\) is a linear rescaling.

Example 7.6

Let \(U\) be a random variable that follows a Uniform(0, 1) distribution.

Let \(Y = 60U\). Sketch a spinner corresponding to the distribution of \(Y\).

\

1. Sketch a plot of the distribution of \(Y\).

1. Let \(Z = U^2\). Sketch a spinner corresponding to the distribution of \(Z\)

1. Sketch a plot of the distribution of \(Z\). (A very rough sketch is fine for now, but be sure to determine if the density would be higher near 0, near 0.5, or near 1.)