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Chapter 6: Problem 10

The joint probability density function of \(X\) and \(Y\) is given by $$f(x, y)=e^{-(x+y)} \quad 0 \leq x<\infty, 0 \leq y<\infty$$ Find (a) \(P\\{X< Y\\}\) and (b) \(P\\{X< a\\}\)

Short Answer

Expert verified
The probability P{X < Y} is \(\dfrac{1}{2}\) and the probability P{X < a} is \(1 - e^{-a}\).

Step by step solution

01

Part (a): Integrate the PDF for X < Y

(1) First, we want to identify the region where X < Y. Since both X and Y are non-negative (0 鈮� X, Y < 鈭�), this region includes the points above the line Y = X in the x-y plane. Graphically, this region will be a right triangle with one vertex at the origin and the other two vertices on the positive X and Y axes. (2) Now we need to integrate the pdf over this region. To do this, we'll start with a double integral over the joint pdf: \( \int\limits_{0}^{\infty}\int\limits_{x}^{\infty} e^{-(x+y)} \, dy \, dx \) (3) First integrate over the Y variable. The integral becomes: \( \int\limits_{0}^{\infty} \left[-e^{-(x+y)}\right]_x^\infty \, dx \) (4) When we evaluate the integrated expression at the limits, we get: \( \int\limits_{0}^{\infty} e^{-2x} \, dx \) (5) Finally, integrate over the X variable and evaluate the limits: \( \left[-\dfrac{1}{2}e^{-2x}\right]_0^\infty = \dfrac{1}{2} \) So, the probability P{X < Y} is 1/2.
02

Part (b): Integrate the PDF for X < a

(1) The region where X < a can be represented as all the points to the left of the vertical line X = a in the x-y plane. Since 0 鈮� X, Y < 鈭�, we need to integrate the pdf over the rectangle with vertices (0, 0), (a, 0), (0, 鈭�), and (a, 鈭�). (2) Now we need to integrate the pdf over this region. To do this, we'll start with a double integral over the joint pdf: \( \int\limits_{0}^{a}\int\limits_{0}^{\infty} e^{-(x+y)} \, dy \, dx \) (3) First integrate the Y variable. The integral becomes: \( \int\limits_{0}^{a} \left[-e^{-(x+y)}\right]_0^\infty \, dx = \int\limits_{0}^{a} e^{-x} - 0 \, dx \) (4) Integrate over the X variable and evaluate the limits: \( \left[-e^{-x}\right]_0^a = 1 - e^{-a} \) So, the probability P{X < a} is \(1 - e^{-a}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Density Function (PDF)
A Probability Density Function (PDF) is a function that describes how the values of a continuous random variable are distributed. For any two points on the random variable's possible spectrum, the area under the PDF curve between these points provides the probability that the variable falls within that interval.

The PDF for continuous random variables is analogous to the probability mass function for discrete random variables, but instead of probabilities for specific values, we talk about probabilities over intervals since the probability at a single point for a continuous variable is effectively zero. A key property of a PDF is that the total area under the curve integrates to 1, representing the certainty that the random variable will take on a value within the range of possible values.

In the given problem, the joint PDF of two continuous random variables, X and Y, is expressed mathematically as \(f(x, y) = e^{-(x+y)}\) for x and y greater than or equal to zero. This function describes the probability density across the two-dimensional space of values that X and Y can simultaneously take.
Integrating PDFs
Integrating a PDF is a fundamental operation in probability theory used to find the probability that a random variable falls within a certain range. The integral of the PDF over a specified range gives this probability.

When dealing with joint PDFs for two variables, like in the exercise, the process involves integrating over a region in two-dimensional space. Double integrals are used, which can be visualized as summing up infinitesimal rectangles, each representing the probability over an infinitesimally small region. The result gives the total probability of the region considered.

For instance, to find the probability that \(X < Y\), we integrate the PDF over the region in the plane where \(X < Y\). This process requires setting up a double integral that respects the boundary conditions of the problem, ensuring that the integration is only performed over the relevant region where the inequality is true.
Probability of Events
The probability of events, in a continuous setting, is calculated by integrating the PDF over the region corresponding to the event. An event is defined as an outcome or a set of outcomes of a random experiment.

In our exercise, calculating the probability that \(X
Similarly, when calculating the probability that \(X
Double Integrals
Double integrals are a tool used in multivariable calculus to calculate the accumulated value over a two-dimensional area. In the context of probability, we use double integrals to integrate over a joint PDF to find the probability of bivariate events.

The integral is performed sequentially, first with respect to one variable while holding the other constant, and then with respect to the second variable. When performing double integrals, the order of integration may be switched if it simplifies the calculations, but care must be taken to properly transform the limits of integration.

In the provided solution, the use of double integrals allows us to find the probabilities \(P\{X

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Most popular questions from this chapter

Consider a sequence of independent Bernoulli trials, each of which is a success with probability \(p .\) Let \(X_{1}\) be the number of failures preceding the first success, and let \(X_{2}\) be the number of failures between the first two successes. Find the joint mass function of \(X_{1}\) and \(X_{2}.\)

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