Question

Let \(\mathrm{X}\) and \(\mathrm{Y}\) be jointly distributed with density function $$ \begin{array}{rlrl} \mathrm{f}(\mathrm{x}, \mathrm{y})= & 1 & 0<\mathrm{x}<1 \\ & & 0<\mathrm{y}<1 \\ & 0 & & \text { otherwise. } \end{array} $$ $$ \text { Find } \quad F(\lambda \mid X>Y)=\operatorname{Pr}(X \leq \lambda \mid X>Y) \text { . } $$

Short answer

The conditional probability of the event X≤λ, given that X>Y is: $$\operatorname{Pr}(X \leq \lambda \mid X>Y)= \frac{4\lambda}{3}$$

Step-by-step solution

Identify the region where X>Y.

We know that 0 < x < 1 and 0 < y < 1. To find the region where X>Y, we look for pairs of x and y such that x > y, which in our case forms a triangular region with vertices (0, 0), (1, 0), and (1, 1).

Find the joint probability of the event X≤λ and X>Y

For this step, we need to find the probability when X≤λ and X>Y. The integration limits will be determined by the triangular region we found. We'll perform a double integral over that region, up to the value of x≤λ: $$\begin{aligned} \operatorname{Pr}(X \leq \lambda \mid X>Y)= & \frac{\operatorname{Pr}(X \leq \lambda, X>Y)}{\operatorname{Pr}(X>Y)} \\ = & \frac{\int_{0}^{\frac{1}{2}} \int_{x}^{\lambda} \mathrm{f}(x, y) \,\mathrm{d}y\,\mathrm{d}x + \int_{\frac{1}{2}}^{1} \int_{x}^{\lambda} \mathrm{f}(x, y) \,\mathrm{d}y\,\mathrm{d}x}{\int_{0}^{1} \int_{0}^{x} \mathrm{f}(x, y) \,\mathrm{d}y\,\mathrm{d}x} \end{aligned}$$

Evaluate the integrals

Since the joint PDF, f(x,y) is 1 when 0Y)= & \frac{\int_{0}^{\frac{1}{2}} \int_{x}^{\lambda} 1 \,\mathrm{d}y\,\mathrm{d}x + \int_{\frac{1}{2}}^{1} \int_{x}^{\lambda} 1 \,\mathrm{d}y\,\mathrm{d}x}{\int_{0}^{1} \int_{0}^{x} 1 \,\mathrm{d}y\,\mathrm{d}x} \\ = & \frac{\int_{0}^{\frac{1}{2}} (\lambda - x)\,\mathrm{d}x + \int_{\frac{1}{2}}^{1} (\lambda - x)\,\mathrm{d}x}{\int_{0}^{1} x\,\mathrm{d}x} \end{aligned}$$ Now, evaluate each integral: $$\begin{aligned} \operatorname{Pr}(X \leq \lambda \mid X>Y)= & \frac{\left[\frac{\lambda}{2}x^2- \frac{1}{3}x^3\right]_0^{\frac{1}{2}} + \left[\frac{\lambda}{2}x^2- \frac{1}{3}x^3\right]_{\frac{1}{2}}^1}{ \left[\frac{x^2}{2}\right]_0^1} \\ = & \frac{\frac{2\lambda}{3}}{\frac{1}{2}} \end{aligned}$$

Simplify the expression

Simplify the expression for the conditional probability: $$\operatorname{Pr}(X \leq \lambda \mid X>Y)= \frac{4\lambda}{3}$$ This is the final expression for the conditional probability of the event X≤λ, given that X>Y.

Key concepts

Joint Probability Distribution

A joint probability distribution provides the probability of outcomes for two random variables occurring simultaneously. For random variables \( X \) and \( Y \), their joint distribution can be described using a probability density function (PDF). This PDF, \( f(x, y) \), informs us about the likelihood of \( X \) taking a particular value \( x \) at the same time \( Y \) takes a value \( y \).

In the exercise, \( X \) and \( Y \) are jointly distributed with a given PDF. This PDF is 1 for values of \( x \) and \( y \) within the range 0 to 1, indicating that outcomes within this range are equally likely. Outside of this range, the probability is 0. This essentially forms a uniform distribution over the unit square in the \( xy \)-plane.

Integration in Probability

In probability, integration is a tool we often use to find probabilities over continuous distributions. Through integration, we aggregate the probabilities assigned by a density function over a specified range. This gives us the total probability over a region of interest.

When dealing with joint probability distributions, like in our exercise, determining the probability of events across multiple variables simultaneously may involve integrating the joint PDF over the relevant section of the plane. This provides us with cumulative probabilities, useful for calculating distributions of derived conditions like \( X > Y \). The integral effectively sums all probabilities where pairs \( (x, y) \) satisfy given constraints.

Probability Density Function

A probability density function (PDF) is a function that specifies the probability of a variable taking on a particular value. However, for continuous variables, the probability of any single point is 0, so the PDF is used to find probabilities over intervals.

In the scenario of jointly distributed variables \( X \) and \( Y \), their PDF, \( f(x, y) \), describes the likelihood of any specific pair \( (x, y) \). For example, our function \( f(x, y) = 1 \) for certain intervals creates a uniform distribution, indicating equal probability across that area. To find specific probabilities, we look at the area under the curve described by this function over our region of interest.

Double Integral in Probability

Double integrals extend the concept of single integration to two dimensions, which is useful when working with jointly distributed random variables. They allow us to calculate the total probability over a two-dimensional region by integrating the joint PDF over that area.

In the exercise, to find conditional probabilities such as \( \operatorname{Pr}(X \leq \lambda \mid X > Y) \), we used double integrals. The numerator required computing a double integral over the region defined by \( X \leq \lambda \) and \( X > Y \). This involves determining the integration boundaries based on these conditions and evaluating the integral to find the probability over the given area. Double integrals streamline the process of managing boundaries and evaluating PDF contributions to the overall probability.