Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 15
The joint density of \(X\) and \(Y\) is given by
$$f(x, y)=\frac{e^{-y}}{y}, \quad 0
Short Answer
Expert verified
The conditional expectation \(E[X^2 | Y=y]\) is \(\frac{y^2}{3}\).
Step by step solution
01
Integrate with respect to x: Because the integrand does not contain \(x\), we can treat it as a constant and integrate directly: $$ f_Y(y) = \left[ \frac{e^{-y}x}{y} \right]_0^y = \frac{e^{-y}y}{y} - \frac{e^{-y}(0)}{y} = e^{-y} $$ #Step 2: Find the conditional density \(f(x | Y=y)\)# Using the joint density function, \(f(x, y)\), and the marginal density of \(Y\), \(f_Y(y)\), we can find the conditional density function \(f(x | Y=y)\) as follows: $$ f(x | Y=y) = \frac{f(x, y)}{f_Y(y)} $$ Substitute both the joint and marginal density functions to get the conditional density: $$ f(x | Y=y) = \frac{\frac{e^{-y}}{y}}{e^{-y}} = \frac{1}{y} $$ #Step 3: Compute the conditional expectation \(E[X^2|Y=y]\)# We can now compute the conditional expectation according to its definition: $$ E[X^2 | Y=y] = \int_{-\infty}^{\infty} x^2 f(x | Y=y) \mathrm{d}x $$ Our range for \(x\) is still between \(0\) and \(y\), and the conditional density is given by \(\frac{1}{y}\): $$ E[X^2 | Y=y] = \int_0^y x^2 \frac{1}{y} \mathrm{d}x $$
Integrate with respect to x:
Integrate this expression and substitute the limits:
$$
E[X^2 | Y=y] = \frac{1}{y} \int_0^y x^2 \mathrm{d}x = \frac{1}{y} \left[\frac{x^3}{3} \right]_0^y = \frac{1}{y}\left(\frac{y^3}{3}\right) - \frac{1}{y}\left(\frac{0^3}{3}\right) = \frac{y^2}{3}
$$
Therefore, the conditional expectation \(E[X^2 | Y=y] = \frac{y^2}{3}\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Joint Density
Joint density provides a way to describe the probability of two continuous random variables, say \(X\) and \(Y\), occurring together. It is denoted by the function \(f(x, y)\). The joint density function must meet two key conditions:
\(f(x, y) \geq 0\) for all \(x\) and \(y\), and the integral over all possible values of \(x\) and \(y\) should equal 1, i.e., \(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \, dx \, dy = 1\).
This means you're essentially looking at a surface above the \(XY\)-plane. The height of this surface at any point \((x, y)\) tells you how likely these particular values of \(X\) and \(Y\) are to occur together.
In the given exercise, the joint density function is \(f(x, y) = \frac{e^{-y}}{y}\). It applies only for \(0 < x < y\) and \(0 < y < \infty\). This indicates the relationship and dependency between \(X\) and \(Y\) as we're trying to find the behavior of \(X\) given \(Y\).
\(f(x, y) \geq 0\) for all \(x\) and \(y\), and the integral over all possible values of \(x\) and \(y\) should equal 1, i.e., \(\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x, y) \, dx \, dy = 1\).
This means you're essentially looking at a surface above the \(XY\)-plane. The height of this surface at any point \((x, y)\) tells you how likely these particular values of \(X\) and \(Y\) are to occur together.
In the given exercise, the joint density function is \(f(x, y) = \frac{e^{-y}}{y}\). It applies only for \(0 < x < y\) and \(0 < y < \infty\). This indicates the relationship and dependency between \(X\) and \(Y\) as we're trying to find the behavior of \(X\) given \(Y\).
Marginal Density
Marginal density focuses on the probability distribution of a subset of a collection of random variables. In simpler terms, it allows us to understand the probability distribution of one variable without considering the others.
To get the marginal density of a variable like \(Y\), you integrate the joint density over all values of the other variable, \(X\). For example:
\[ f_Y(y) = \int_{-\infty}^{\infty} f(x, y) \mathrm{d}x. \]
In the exercise, we find that \( f_Y(y) = e^{-y} \). This extraction process isolates \(Y\) and strips away the dependency on \(X\).
A good way to picture marginal density is by imagining you're flattening a 3D graph into two dimensions to focus solely on one axis.
To get the marginal density of a variable like \(Y\), you integrate the joint density over all values of the other variable, \(X\). For example:
\[ f_Y(y) = \int_{-\infty}^{\infty} f(x, y) \mathrm{d}x. \]
In the exercise, we find that \( f_Y(y) = e^{-y} \). This extraction process isolates \(Y\) and strips away the dependency on \(X\).
A good way to picture marginal density is by imagining you're flattening a 3D graph into two dimensions to focus solely on one axis.
Conditional Density
Conditional density comes into play when we want to find the probability of \(X\) given a certain value of \(Y\). We mathematically represent it as \(f(x | Y=y)\). It's calculated using the joint and marginal densities with the formula:
\[ f(x | Y=y) = \frac{f(x, y)}{f_Y(y)}. \]
Using this, you're essentially narrowing down your focus to a specific "slice" of the joint distribution for a particular \(y\).
For our problem, substituting the appropriate functions, we found \(f(x | Y=y) = \frac{1}{y}\). This implies that given \(Y\), \(X\) is uniformly distributed over its range \(0 < x < y\). The conditional density tells us how the distribution of \(X\) changes when \(Y\) is known.
\[ f(x | Y=y) = \frac{f(x, y)}{f_Y(y)}. \]
Using this, you're essentially narrowing down your focus to a specific "slice" of the joint distribution for a particular \(y\).
For our problem, substituting the appropriate functions, we found \(f(x | Y=y) = \frac{1}{y}\). This implies that given \(Y\), \(X\) is uniformly distributed over its range \(0 < x < y\). The conditional density tells us how the distribution of \(X\) changes when \(Y\) is known.
Probability Models
Probability models help us make sense of randomness and uncertainty in complex phenomena. They use mathematical frameworks to predict the likelihood of different outcomes based on known data or inherent randomness.
In this exercise, we're employing a model that describes how \(X\) and \(Y\) are related through their joint densities. By understanding their marginals and conditionals, we build a complete stochastic story for these variables.
Probability models start with assumptions about the underlying distributions, which affect how you interpret data from real-world scenarios. They offer ways several types of such assumptions
These models are essential for everything from theoretical studies to practical applications in engineering, economics, biology, social sciences, and many more fields of research.
In this exercise, we're employing a model that describes how \(X\) and \(Y\) are related through their joint densities. By understanding their marginals and conditionals, we build a complete stochastic story for these variables.
Probability models start with assumptions about the underlying distributions, which affect how you interpret data from real-world scenarios. They offer ways several types of such assumptions
- Continuous Models: Like ours, dealing with continuous random variables.
- Discrete Models: Where outcomes take on countable values.
- Mixed Models: Incorporate both continuous and discrete variables.
These models are essential for everything from theoretical studies to practical applications in engineering, economics, biology, social sciences, and many more fields of research.
