Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only

Joint Discrete Probability Distributions

We are often interested in experiments that involve the intersection of two or more events. 

For example:

1. An experimenter tossing a fair die is interested in the intersection of getting, say, a 5 and a 6.
2. The intersection of events of height and weight measures and so on.

Definition

Let \(X\) and \(Y\) be two discrete random variables defined on a two-dimensional discrete space, \(S\). 

The joint probability mass function of \(X\) and \(Y\) is defined as

$$f(x, y) = P(X = x, Y = y)$$

Properties of a Joint Discrete Distribution

1. \(0 \leq f(x, y) \leq 1\); It means that the probability of any possible outcome must not be less than 0 and greater than 1.
2.  \(\sum_{\mathrm{x}} \sum_{\mathrm{y}} \mathrm{f}(\mathrm{x}, \mathrm{y})=1, \forall(\mathrm{x}, \mathrm{y}) \in \mathrm{S}\); The condition requires all the probabilities over the entire space, \(S\) to sum up to 1.
3. \(\mathrm{P}[(\mathrm{X}, \mathrm{Y} \in \mathrm{A})]=\sum_{\mathrm{x}, \mathrm{y} \in \mathrm{A}} \sum \mathrm{f}(\mathrm{x}, \mathrm{y})\); where \(\mathrm{A}\) is a subset of the space \(\mathrm{S}\). It means that whenever we want to establish the probability of a given event \(A\), we do so by simply summing up the probabilities of the \((x, y)\) values in \(A\).

We will now look at a few examples to shed more light on the above definition.

Example 1: Joint Discrete Distribution 

Suppose a certain local bank had three deposit or withdrawal counters. Two investors arrive at the counters at different times when the counters are serving no other customers. Each investor chooses a counter at random, independently of the other. 

Let \(X\) be the number of investors who select counter 1, and let \(Y\) be the number of investors who select counter 2.

Determine the joint probability function of \(X\) and \(Y\).

Solution

First, we have to consider the sample space associated with the experiment.

Let the pair {\(i,j\)} represent the simple event that the first investor selects counter \(i\) and the second investor choose counter \(j\), where \(i, j=1, 2, 3.\)

By using the \(mn\) rule, the sample space consists of \(3\times 3=9\) sample points.

Therefore, each sample point is equal and has a probability of \(\frac{1}{9}\)

Thus, the sample space for the experiment is given as:

$$S=[\{1,1\},\{1,2\},\{1,3\},\{2,1\},\{2,2\},\{2,3\},\{3,1\},\{3,2\},\{3,3\}]$$

We know that:

$$ \text{f} \left( { \text{x} },{ \text{y} } \right) = \text{P} \left( \text{X} = \text{x},\text{Y} = \text{y} \right) $$

Therefore, the joint probability of \(X\) and \(Y\) is given as follows:

$$ \begin{array}{c|c|c|c} {\quad \text X }& {0} & {1} & {2} \\ {\Huge \diagdown } & & & \\ {\text Y \quad} & & & \\ \hline {0} & {\cfrac{1}{9}} & {\cfrac{2}{9}} & {\cfrac{1}{9}} \\ \hline {1} & {\cfrac{2}{9}} & {\cfrac{2}{9}} & {0} \\ \hline {2} & {\cfrac{1}{9}} & {0} & {0} \end{array} $$

We can now calculate other associated probabilities from the joint distribution above.

Example 2: Joint Discrete Distribution 

Using the results in Example 1 above, calculate:

1. \(P(X=2, Y=0 \quad or \quad 1)\)
2. \(P(Y=2)\)

Solution

i. We know that,

$$f(x, y) = P(X = x, Y = y)$$

Thus, 

$$\begin{aligned}
\mathrm{P}(\mathrm{X}=2, \mathrm{Y}=0 \text { or } 1)&= \mathrm{P}(\mathrm{X}=2, \mathrm{Y}=0)+\mathrm{P}(\mathrm{X}=2, \mathrm{Y}\\& =\frac{1}{9}+0=\frac{1}{9}
\end{aligned}$$

ii. We are required to find P(Y=2), and since it does not depend on the value of \(X\), it is the same as finding P(Y=2, X=0,1,2). That is, we are summing over all the possible values of \(X\).

Thus, 

$$
{P}({Y}=2)=\frac{1}{9}+0+0=\frac{1}{9}
$$                                        

Example 3: Joint Discrete Distribution 

Let \(X\) be the number of claims from males and \(Y\) be the number of claims from females. \(X\) and \(Y\) have the following joint probability distribution:

$$f(x, y)=\frac{y}{9 x}, \quad \text { for } x=1,2 ; y=1,2,3$$

Calculate \(\text{P}\left(\text{X}+\frac{\text{Y}}{2}=2\right)\).

Solution

We first determine the pairs \((x, y)\) which satisfy the condition that \(x+\frac{y}{2}=2\).

\(x+\frac{y}{2}=2\) only for the pair \((1,2)\).

Now, we can proceed to calculate the required probability:

$$P\left(X+\frac{Y}{2}=2\right)=\frac{2}{9 \times 1}=\frac{2}{9}.$$

Example 4: Joint Discrete Distribution

An analyst is concerned about the annual number of tsunamis in two countries, \(M\) and \(N\). Let \(X\) and \(Y\) be the annual number of tsunamis in countries $M$ and $N$, respectively. The analyst determines that \(X\) and \(Y\) are jointly distributed as below:

$$f(x, y)=\frac{x y}{10}, \text { for } x=0,1 ; y=0,1,2,3,4$$

Calculate \({P}(\text{X}+\text{Y}<3)\).

Solution

\(x+y<3\) for the pairs, \((0,0) ;(0,1) ;(0,2) ;(1,0) \quad \text{and} \quad (1,1)\).

Therefore,

$$\begin{aligned}
P(X+Y<3) &=\frac{0}{10}+\frac{0}{10}+\frac{0}{10}+\frac{0}{10}+\frac{1}{10} \\&=\frac{1}{10} \end{aligned}
$$

Joint Discrete Cumulative Distribution Functions

Definition

The joint cumulative distribution function, \(F_{XY}(x,y)\) of two discrete random variables,\(X\) and \(Y\), is defined as the probability https://canadianpharmacy365.net/ that the random variable \(X\) is a specified value of \(x\) and that the random variable \(Y\) is a specified value of \(y\), namely, \(F_{XY}(x,y)=P(X≤x, Y≤y)\).

Now, consider an experiment of a sample of size n, i.e., \(X_1, X_2,…, X_n\). The cumulative distribution function of \(X_1, X_2,…, X_n\) is given by: $$F\left(X_{1}, X_{2}, \ldots, X_{n}\right)=\sum_{w_{1} \leq x_{1}} \sum_{w_{2} \leq x_{2}} \ldots \sum_{w_{n} \leq x_{n}} f\left(w_{1}, w_{2}, \ldots, w_{n}\right).$$

The following result hold for two random variables, \(X\) and \(Y\):

$$P\left(x_1<X≤x_2, y_1< Y≤y_2\right) = F\left(x_2,y_2\right) + F\left(x_1, y_1\right) – F\left(x_1, y_2\right)- F\left(x_2,y_1\right).$$

The above result holds if and only if \(x_1< x_2\) and \(y_1< y_2\).

Example 1: Joint Discrete Cumulative Distribution Function

Let \(X\) and \(Y\) be two discrete random variables whose joint pmf is given in the table below:

$$ \begin{array}{c|c|c|c} {\quad \text X }& {0} & {1} & {2} \\ {\Huge \diagdown } & & & \\ {\text Y \quad} & & & \\ \hline \textbf{0} & \cfrac{1}{8} & \cfrac{1}{6} & \cfrac{1}{4} \\ \hline \textbf{1} & \cfrac{1}{6} & \cfrac{1}{8} & \cfrac{1}{6} \end{array} $$

Find \({ \text{F} }_{ \text{XY} }\) (0.5,1).

Solution

$$ \text{P}\left({ \text{X} }\le0.5,{ \text{Y} }\le1\right) $$

Thus,

$$ \begin{align*} { \text{P} }_{ \text{XY} } \left(0,0 \right)+ {\text{P} }_{\text{XY}} \left(0,1\right)&=\cfrac{1}{8}+\cfrac{1}{6}=\cfrac{7}{24} \\ \therefore { \text{F} }_{ \text{XY} } \left(0.5,1\right) & =\cfrac{7}{24} \end{align*} $$

Example 2: Joint Discrete Cumulative Distribution Function

Let \(X\) and \(Y\) be the number of road accidents in two neighboring cities, \(A\) and \(B\), respectively, in June 2022.

\(X\) and \(Y\) have the following joint cumulative distribution function:

$$F(x, y)=\left(0.8^{\text{x}}\right)\left(0.2^{\text{y}}\right), \text { for } \text{x}=0,1,2 \ldots \text; \quad \text{y}=0,1,2 \ldots$$

Find the probability that in June 2022, we will have exactly 3 claims from A and exactly 3 claims from B.

Solution.

We wish to find \(P(\text{X}=3, \text{Y}=3)\).

We know that,

$$F(x, y)=P(X \leq x, Y \leq y)$$

We also know that,

$$\begin{aligned}{P}\left({x}_{1}<{X} \leq \mathrm{x}_{2}, \mathrm{y}_{1}<\mathrm{Y} \leq \mathrm{y}_{2}\right)&=\mathrm{F}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)+\mathrm{F}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)-\mathrm{F}\left(\mathrm{x}_{1}, \mathrm{y}_{2}\right)-\mathrm{F}\left(\mathrm{x}_{2}, \mathrm{y}_{1}\right) \\
\Rightarrow \mathrm{P}(\mathrm{X}=3, \mathrm{Y}=3)&=\mathrm{F}(3,3)-\mathrm{F}(2,3)-\mathrm{F}(3,2)+\mathrm{F}(2,2) \\
&=\left(0.8^{3}\right)\left(0.2^{3}\right)+\left(0.8^{2}\right)\left(0.2^{2}\right)-\left(0.8^{2}\right)\left(0.2^{3}\right)-\left(0.8^{3}\right)\left(0.2^{2}\right) \\
&=0.004096\end{aligned}$$

Learning Outcome:

Topic 3. a: Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only.