91Ó°ÊÓ

An exponential distribution is often used to model the time between events in a Poisson process. It has a constant failure rate and is characterized by the parameter \(\lambda\), which is the rate parameter. In practice, this means if \(\lambda\) is larger, the waiting time for an event is shorter. When the mean waiting time is one, \(\lambda = 1\). For the exponential distribution, the probability density function (pdf) is given by: \[ f(x) = \lambda e^{-\lambda x}, \quad x \geq 0 \] The exponential distribution is memoryless, meaning that the probability of an event occurring in the future is independent of the past. This property is unique for exponential distributions and simplifies many calculations.

• The pdf decreases exponentially, illustrating the rapid decline in probability as the waiting time increases.
• It is a continuous distribution, supporting that the time variable is continuous rather than discrete.

Marginal density refers to the probability distribution of a subset of a collection of random variables. In other words, it is what you get when you look at one variable in isolation. To find the marginal density of a variable, you integrate out the other variables from the joint probability density function (pdf). For example: To find the marginal density of \(X\), we integrate the joint pdf \(f(x,y)\) over all values of \(y\) from \(x\) to infinity: \[ f_X(x) = \int_x^\infty f(x, y) \, dy \] This process simplifies the consideration from two variables \(X\) and \(Y\) to just one, \(X\).

• Makes calculations more manageable by focusing on less complex distributions.
• Essential for deriving properties and probabilities involving individual variables from joint behavior.

Conditional probability quantifies the probability of an event given that another event has occurred. It is particularly helpful in deriving probabilities when additional information is available. In the context of two random variables, \(X\) and \(Y\), the conditional probability density function \(f_{Y|X}(y|x)\) is calculated using: \[ f_{Y|X}(y|x) = \frac{f(x, y)}{f_X(x)} \] This expression shows that the conditional density is derived by dividing the joint pdf by the marginal pdf of \(X\). By doing so, it adjusts the probability distribution of \(Y\) to reflect the new "condition" that \(X = x\).

• Enables understanding of the relationship between \(X\) and \(Y\) beyond joint properties.
• Essential for calculating probabilities under given conditions.

This probability is dynamic as it often involves changes with different conditions that make problem-solving targeted and efficient.

Two random variables \(X\) and \(Y\) are independent if the occurrence of one does not affect the probability of the other. In terms of joint probability densities, \(X\) and \(Y\) are independent if the joint density \(f(x, y)\) equals the product of their marginal densities \(f_X(x)\) and \(f_Y(y)\): \[ f(x, y) = f_X(x) \cdot f_Y(y) \] If this condition holds, it implies that knowing the outcome of one variable provides no information about the other. Independence greatly simplifies calculations because it allows the separation of variables in probability expressions.

• Leads to simplified analysis and computation in probability problems.
• Aids in understanding the interaction (or lack thereof) between variables by comparing joint behavior against combined individual behavior.

In this exercise, \(X\) and \(Y\) are found to not be independent, because their joint pdf does not factor into the product of individual marginal pdfs. This means that one variable influences the distribution of the other, at least partially.