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Magazine Chapter 5: Problem 10
Let \(X\) and \(Y\) be independent exponential random variables with respective rates \(\lambda\) and \(\mu\). Let \(M=\min (X, Y)\). Find (a) \(E[M X \mid M=X]\) (b) \(E[M X \mid M=Y]\), (c) \(\operatorname{Cov}(X, M)\)
Short Answer
Expert verified
The short answers for the given question are:
(a) \(E[M X \mid M = X] = \frac{2}{(\lambda + \mu)^2}\)
(b) \(E[M X \mid M = Y] = \frac{2\lambda}{\mu(\lambda + \mu)^2}\)
(c) \(\operatorname{Cov}(X, M) = \frac{\mu}{\lambda(\lambda + \mu)^2}\)
Step by step solution
01
(a) Calculate the joint probability density function (pdf) of X and Y
Since X and Y are independent exponential random variables with rates 位 and 渭 respectively, their joint probability density function (pdf) is the product of their individual pdfs:
\(f(x, y) = \lambda e^{-\lambda x} \mu e^{-\mu y}\) for \(x, y \geq 0\).
02
(a) Calculate the conditional pdf of X, given M = X
We have to condition on the event that the minimum of X and Y is X, which happens when X < Y. So, we calculate the conditional pdf f_X(x | M = X) as follows:
\(f_{X|M}(x | M=X) = \frac{f(x, y > x)}{P(M = X)}\) = \(\frac{\int_x^\infty \lambda e^{-\lambda x} \mu e^{-\mu y} dy}{P(M = X)}\)
Now we need to calculate \(P(M = X)\), which is given by:
\(P(M = X) = \int_0^\infty P(X < Y | X = x) f_X(x) dx = \int_0^\infty e^{-\mu x} \lambda e^{-\lambda x} dx = \frac{\lambda}{\lambda + \mu}\)
Hence, the conditional pdf is:
\(f_{X|M}(x | M=X) = (\lambda + \mu) e^{-x(\lambda + \mu)}\) for \(x \geq 0\).
03
(a) Calculate \(E[M X | M = X]\)
Now we can find the conditional expectation as follows:
\(E[M X | M = X] = \int_0^\infty x^2 (\lambda + \mu) e^{-x(\lambda + \mu)} dx\)
Applying integration by parts, we get:
\(E[M X | M = X] = \frac{2}{(\lambda + \mu)^2}\)
04
(b) Calculate \(E[M X | M = Y]\)
Since \(M = Y\) in this case, we need to find \(E[Y X | M = Y]\). First, we find the conditional pdf of Y, given M = Y, similar to what we did for X:
\(f_{Y|M}(y | M=Y) = \frac{f(x > y, y)}{P(M = Y)}\) = \(\frac{\int_y^\infty \lambda e^{-\lambda x} \mu e^{-\mu y} dx}{P(M = Y)}\)
Notice that \(P(M = Y) = 1 - P(M = X) = \frac{\mu}{\lambda + \mu}\).
Therefore, the conditional pdf is:
\(f_{Y|M}(y | M=Y) = (\lambda + \mu) e^{-y(\lambda + \mu)}\) for \(y \geq 0\).
Now we can find the conditional expectation:
\(E[Y X | M = Y] = \int_0^\infty\int_0^y xy (\lambda + \mu)^2 e^{-x(\lambda + \mu)} e^{-y(\lambda + \mu)} dx dy\)
\(E[Y X | M = Y] = \frac{\lambda}{\mu} E[M X | M = X] = \frac{2\lambda}{\mu(\lambda + \mu)^2}\)
05
(c) Calculate the Covariance Cov(X, M)
To find the covariance Cov(X, M), we first need to find the marginal pdf of M and the joint pdf of (X,M), and then we can compute the required expectations:
\(E[X] = \int_0^\infty x \lambda e^{-\lambda x} dx = \frac{1}{\lambda}\)
\(E[M] = P(M = X)E[M | M = X] + P(M = Y)E[M | M = Y] = \frac{1}{\lambda + \mu}\)
Where \(E[M | M = X] = \int_0^\infty x (\lambda + \mu) e^{-x(\lambda + \mu)} dx = \frac{1}{\lambda + \mu}\) and \(E[M | M = Y] = \frac{1}{\lambda + \mu}\).
Now we calculate the joint expectations:
\(E[XM] = E[X M | M = X] P(M = X) + E[X M | M = Y] P(M = Y) = \frac{2}{(\lambda + \mu)^2} \frac{\lambda}{\lambda + \mu} + \frac{2\lambda}{(\lambda + \mu)^2 \mu} \frac{\mu}{\lambda + \mu}\)
\(E[XM] = \frac{1}{(\lambda + \mu)^2}\)
Finally, we find the covariance:
\(\operatorname{Cov}(X, M) = E[XM] - E[X]E[M] = \frac{1}{(\lambda + \mu)^2} - \frac{1}{\lambda(\lambda + \mu)} = \frac{\mu}{\lambda(\lambda + \mu)^2}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Probability Density Function
Understanding the probability density function (pdf) is essential for grasping various concepts in probability and statistics. The pdf is a function that describes the relative likelihood of a continuous random variable to take on a given value. In mathematical terms, for a continuous random variable \(X\), the probability that \(X\) falls within a particular range is given by the integral of its density function over that range.
The pdf is denoted by \(f(x)\) and is such that for any two numbers \(a\) and \(b\) with \(a < b\), the probability that \(X\) lies between \(a\) and \(b\) is \(\int_a^b f(x) dx\). A key property of the pdf is that the total area under the curve of \(f(x)\) over all possible values of \(X\) is 1, symbolizing the fact that the probability of the random variable taking on any value is 100%.
In the exercise context, the independent exponential random variables \(X\) and \(Y\) each have a pdf, and since they are independent, their joint pdf \(f(x, y)\) is the product of their individual pdfs.
The pdf is denoted by \(f(x)\) and is such that for any two numbers \(a\) and \(b\) with \(a < b\), the probability that \(X\) lies between \(a\) and \(b\) is \(\int_a^b f(x) dx\). A key property of the pdf is that the total area under the curve of \(f(x)\) over all possible values of \(X\) is 1, symbolizing the fact that the probability of the random variable taking on any value is 100%.
In the exercise context, the independent exponential random variables \(X\) and \(Y\) each have a pdf, and since they are independent, their joint pdf \(f(x, y)\) is the product of their individual pdfs.
Exponential Random Variables
Exponential random variables are widely used to model the time until an event occurs, such as the time until a radioactive particle decays, or the time between customer arrivals in a queue. The key characteristic of an exponential random variable \(X\) is that it has a memoryless property, which means that the probability of an event occurring in the next interval is independent of how much time has already elapsed.
The probability density function of an exponential random variable with rate \(\lambda\) is given by \(f(x) = \lambda e^{-\lambda x}\) for \(x \geq 0\), where \(1/\lambda\) is the mean of the distribution. In our example, both \(X\) and \(Y\) are exponential random variables, which simplifies the analysis of the minimum of the two, denoted by \(M\), and allows us to explore interesting properties, like the conditional expectations given in the exercise.
The probability density function of an exponential random variable with rate \(\lambda\) is given by \(f(x) = \lambda e^{-\lambda x}\) for \(x \geq 0\), where \(1/\lambda\) is the mean of the distribution. In our example, both \(X\) and \(Y\) are exponential random variables, which simplifies the analysis of the minimum of the two, denoted by \(M\), and allows us to explore interesting properties, like the conditional expectations given in the exercise.
Covariance of Random Variables
Covariance is a measure of how much two random variables change together. If the variables tend to increase and decrease simultaneously, they have positive covariance. Contrarily, if one variable tends to increase when the other decreases, the covariance is negative. If the covariance is zero, this can imply that the variables are independent, but the reverse is not always true; dependent variables can also have zero covariance if their relationship is not linear.
Mathematically, the covariance between two random variables \(X\) and \(Y\) is defined by \(\operatorname{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y]\), where \(E[\cdot]\) denotes the expected value. In the context of our exercise, we are interested in the covariance between \(X\) and the minimum \(M\), which tells us how 'X' changes with the minimum of 'X' and another variable 'Y'. Understanding covariance helps in studying the linear relationship between variables and is crucial for further concepts like correlation and regression analysis.
Mathematically, the covariance between two random variables \(X\) and \(Y\) is defined by \(\operatorname{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y]\), where \(E[\cdot]\) denotes the expected value. In the context of our exercise, we are interested in the covariance between \(X\) and the minimum \(M\), which tells us how 'X' changes with the minimum of 'X' and another variable 'Y'. Understanding covariance helps in studying the linear relationship between variables and is crucial for further concepts like correlation and regression analysis.
