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When considering probability distributions, understanding joint probability is essential. It describes the likelihood of two events occurring simultaneously. For two random variables, like our variables \(X\) and \(Y\) representing vehicle arrivals at two intersections, the joint probability distribution provides information about how these variables might interact with each other.
To compute the value of joint probability, you'll need a joint probability function, such as the one provided in the exercise: \[f(x, y) = \frac{1}{4^{(x+y)}} \cdot \frac{9}{16}\] This function evaluates the probability of different combinations of \(X\) and \(Y\). You can use it to build a probability table for precise analysis, enabling calculation of both expected values and variances for each of the variables.

The expected value of a random variable is like the average or mean, providing an idea of its central tendency. Formally, it is the weighted average of all possible values. For randomly arriving vehicles \(X\) and \(Y\), the expected values \(E(X)\) and \(E(Y)\) can be determined using the formula:

• \(E(X) = \sum x \cdot P(X = x)\)
• \(E(Y) = \sum y \cdot P(Y = y)\)

These sums involve multiplying each possible outcome by its probability and then adding them all together. The provided function allows you to substitute \(x\) and \(y\) values with their associated probabilities to find these expectations. Once calculated, they give insights into the average number of vehicles expected to arrive at each intersection during the period.

Variance gives you an idea of how much the values of a random variable spread out from the expected value. It's crucial for determining the unpredictability of the variable. For our variables \(X\) and \(Y\), variance is calculated using the formula:
\[\operatorname{Var}(X) = E[(X - E(X))^2] = E[X^2] - (E[X])^2\]\[\operatorname{Var}(Y) = E[(Y - E(Y))^2] = E[Y^2] - (E[Y])^2\] These expressions calculate how far values typically lie from the mean, squared for positive numbers, providing a sense of variability. To apply, compute each value's square, calculate its expectation, and utilize the previously found expected values.

In probability and statistics, a random variable is a variable whose possible values are numerical outcomes of a random phenomenon. Unlike deterministic variables, they do not have fixed values, allowing them to model the randomness of real-life situations such as traffic.Random variables can be discrete or continuous, and the exercise here focuses on discrete random variables \(X\) and \(Y\).

• \(X\) and \(Y\) represent the number of arrivals in a given period at distinct intersections.
• The joint probability distribution describes their simultaneous behavior.

Understanding how these random variables work and the impact of their probability distributions is key to analyzing such scenarios, allowing for better predictions and strategic decisions in real-world contexts.