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Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. In the context of the original exercise, several problems involve finding conditional probabilities, such as \(P(X < 1 \mid Y=1)\) and \(P(X^2 + Y^2 < 1 \mid Z=1)\). To understand these, it's crucial to know that conditional probability changes the probability space to reflect known information.
For instance, determining \(P(X < 1 \mid Y = 1)\) changes our attention to only those instances where \(Y\) equals 1. This involves integrating over a narrower region than the whole initial probability space. It often requires adjusting the probability of all relevant events by the probability of the given event. In general, conditional probability is calculated using the formula:

• \(P(A \mid B) = \frac{P(A \cap B)}{P(B)}\)

The expected value is a measure of the central tendency of a random variable. It provides an average or "expected" value when an experiment is repeated a large number of times. This is especially useful in probability, helping us understand the likely outcomes of various random events. For a continuous variable like \(X\), the expected value is computed as:

• \(E(X) = \int x f_X(x) \, dx\)

In the exercise, because of the symmetry of the function over a symmetric region, the expected value \(E(X)\) can be immediately deduced to be 0.
Symmetry plays a crucial role here. This is because the probability distribution is even and symmetric around zero. Therefore, the positive and negative values balance each other out perfectly.

Uniform distribution is a probability distribution where all outcomes are equally likely within a specified range. In the problem, \(Z\) is uniformly distributed given specific conditions. When a variable is uniformly distributed over an interval, it means that the probability density function (pdf) is constant over that interval. For a continuous uniform distribution, this is described mathematically by:

• \(f_X(x) = \frac{1}{b-a}, \quad a \leq x \leq b\)

Here \(b\) and \(a\) are the boundaries of the interval. Uniform distributions are straightforward because they suggest all values within the range are equally probable. For \(Z\) in the original problem, knowing \(X=1\) and \(Y=1\) positions \(Z\) uniformly between \(0\) and \(4\). This simplification is helpful in reducing complex probability problems to basic calculations.

Integration is a powerful tool in probability, especially for continuous random variables. It allows us to calculate probabilities, expected values, and variances for these variables by evaluating continuous probability density functions (pdfs) over certain intervals.
In the step-by-step solution to the exercise, integration is used to find the constant \(c\) that normalizes the joint pdf so that the total probability equals one. It also helps calculate several probabilities of interest, like \(P(X^2 + Y^2 < 2)\).
Integration essentially sums up all infinitesimally small probabilities over a particular region.
The process follows these general steps:

• Identify the region of integration, which is the range where the random variable or variables are defined.
• Set up the integral using the given pdf over these regions.
• Evaluate the integral to find the required probability or expected value.

Integration facilitates transforming complex probability geometry into solvable mathematical expressions.