Problem 53 - A First Course in Probability

Question

If \(X\) and \(Y\) are independent random variables both uniformly distributed over \((0,1),\) find the joint density function of \(R=\sqrt{X^{2}+Y^{2}}, \Theta=\tan ^{-1} Y / X.\)

Step-by-Step Solution

Answer

\( f_{R,\Theta}(r,\theta) = r \) for \( 0<\theta<\pi/2 \), \( 0

Step 1: Transformation

X, Y ~ Uniform(0,1) independent. \( R=\sqrt{X^2+Y^2} \), \( \Theta=\tan^{-1}(Y/X) \). Inverse: \( X=R\cos\Theta, Y=R\sin\Theta \).

Step 2: Jacobian

\( f_{X,Y} = 1 \) on \( (0,1)^2 \). Jacobian \( |J| = r \).

Step 3: Result

\( f_{R,\Theta}(r,\theta) = r \) for \( 0<\theta<\pi/2 \), \( 0

Key Concepts

Independent Random Variables

Understanding the concept of independent random variables is foundational in the study of probability and statistics. Two random variables, X and Y, are considered independent if the occurrence of one does not affect the probability of occurrence of the other. In mathematical terms, they are independent if the probability of both X and Y occurring together is the product of their individual probabilities.

When we have independent random variables, it simplifies our calculations significantly. For instance, finding joint probabilities or joint density functions becomes a matter of multiplying the probability density functions (PDFs) of the individual random variables, provided they are independent. This concept is a cornerstone in solving problems that deal with the combination of different stochastic processes or events, such as the exercise involving the transformation of two uniform random variables to polar coordinates.

Uniform Distribution

Characteristics of Uniform Distribution

Uniform distribution is a type of probability distribution where all outcomes are equally likely to occur within a certain interval. If random variables X and Y follow a uniform distribution over the interval (0,1), it suggests that every value between 0 and 1 is just as likely as any other.

Random variables with uniform distribution are often used to represent idealized situations where each outcome is equally likely, such as a perfectly fair die or lottery. Uniform distributions can be described by their probability density function (PDF), which for the continuous case, is constant within the bounds of the distribution and zero elsewhere.

Probability Density Function

The probability density function (PDF) is used to specify the probability of a random variable falling within a particular range of values, as opposed to taking on any one specific value. A higher PDF value at a certain point or interval means a higher likelihood that the random variable will occur near that point or within that interval.

In the case of a continuous uniform distribution over (0,1), the PDF would be 1 for all values of X within the range and 0 otherwise. It is crucial in calculating probabilities and is heavily relied upon in problem-solving within statistics and probability theory, especially when it comes to continuous random variables, which is what the original exercise entails.

Transformation of Random Variables

Change of Variables Method

Transformation of random variables is a technique used to change the variables in a given problem to new variables which may be more suitable for analysis or finding solutions. This transformation can simplify complex probability problems, especially when changing from Cartesian to polar coordinates as seen in our exercise.

To find the joint density function of the transformed variables, a method called the 'Change of Variables' or 'Jacobian Transformation' is typically used. In our problem, we use the transformation from X and Y, which are independent uniform random variables, to R and Theta in polar coordinates. This change requires calculating the Jacobian, which is the determinant of the matrix of all first-order partial derivatives of the transformation functions. The Jacobian assesses how a point in a plane changes as it is transformed from one coordinate system to another, which in this exercise, aids in determining the joint density of R and Theta.

Previous

Problem 52

Next

Problem 55

Other exercises in this chapter

Problem 51

Derive the distribution of the range of a sample of size 2 from a distribution having density function \(f(x)=\) \(2 x, 0

View solution

Problem 52

Let \(X\) and \(Y\) denote the coordinates of a point uniformly chosen in the circle of radius 1 centered at the origin. That is, their joint density is $$f(x,

View solution

Problem 55

\(X\) and \(Y\) have joint density function $$f(x, y)=\frac{1}{x^{2} y^{2}} \quad x \geq 1, y \geq 1$$ (a) Compute the joint density function of \(U=X Y, V=\) \

View solution

Problem 57

Repeat Problem 6.56 when \(X\) and \(Y\) are independent exponential random variables, each with parameter \(\lambda=1.\)

View solution

Key Concepts

Independent Random Variables

Understanding the concept of independent random variables is foundational in the study of probability and statistics. Two random variables, X and Y, are considered independent if the occurrence of one does not affect the probability of occurrence of the other. In mathematical terms, they are independent if the probability of both X and Y occurring together is the product of their individual probabilities.

When we have independent random variables, it simplifies our calculations significantly. For instance, finding joint probabilities or joint density functions becomes a matter of multiplying the probability density functions (PDFs) of the individual random variables, provided they are independent. This concept is a cornerstone in solving problems that deal with the combination of different stochastic processes or events, such as the exercise involving the transformation of two uniform random variables to polar coordinates.

Uniform Distribution

Characteristics of Uniform Distribution

Uniform distribution is a type of probability distribution where all outcomes are equally likely to occur within a certain interval. If random variables X and Y follow a uniform distribution over the interval (0,1), it suggests that every value between 0 and 1 is just as likely as any other.

Random variables with uniform distribution are often used to represent idealized situations where each outcome is equally likely, such as a perfectly fair die or lottery. Uniform distributions can be described by their probability density function (PDF), which for the continuous case, is constant within the bounds of the distribution and zero elsewhere.

Probability Density Function

The probability density function (PDF) is used to specify the probability of a random variable falling within a particular range of values, as opposed to taking on any one specific value. A higher PDF value at a certain point or interval means a higher likelihood that the random variable will occur near that point or within that interval.

In the case of a continuous uniform distribution over (0,1), the PDF would be 1 for all values of X within the range and 0 otherwise. It is crucial in calculating probabilities and is heavily relied upon in problem-solving within statistics and probability theory, especially when it comes to continuous random variables, which is what the original exercise entails.

Transformation of Random Variables

Change of Variables Method

Transformation of random variables is a technique used to change the variables in a given problem to new variables which may be more suitable for analysis or finding solutions. This transformation can simplify complex probability problems, especially when changing from Cartesian to polar coordinates as seen in our exercise.

To find the joint density function of the transformed variables, a method called the 'Change of Variables' or 'Jacobian Transformation' is typically used. In our problem, we use the transformation from X and Y, which are independent uniform random variables, to R and Theta in polar coordinates. This change requires calculating the Jacobian, which is the determinant of the matrix of all first-order partial derivatives of the transformation functions. The Jacobian assesses how a point in a plane changes as it is transformed from one coordinate system to another, which in this exercise, aids in determining the joint density of R and Theta.