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Chapter 4: Problem 6

Let \(X_{1}, X_{2}, X_{3}\) be a random sample from a distribution of the continuous type having pdf \(f(x)=2 x, 0

Short Answer

Expert verified
For part (a), the probability that the smallest of \(X_1, X_2, X_3\) exceeds the median of the distribution is \([1 - 1/\sqrt{2}]^3\). For part (b), the correlation between \(Y_2\) and \(Y_3\) is the result of the last computation in Step 3.

Step by step solution

01

Part (a) 鈥� Step 1: Finding the Median

The median is defined as the value separating the higher half from the lower half of a data sample. In our case, this implies that the probability of a random variable being less than the median is equal to the probability of being more. Formally, we need to solve for \(m\) in the equation: \(\int_{0}^{m} 2x dx = 1/2\). Solving this, we find \(m=1/\sqrt{2}\).
02

Part (a) 鈥� Step 2: Compute the Required Probability

Now, the required probability is that the smallest of the \(X_i\)'s exceeds the median, or equivalently, that all \(X_i\)'s exceed the median. Hence, this is equal to \([ \int_{1/\sqrt{2}}^{1} 2x dx ]^3 = [1 - 1/\sqrt{2}]^3\).
03

Part (b) 鈥� Step 1: Computing Joint Probability Density Function

As \(Y_1\), \(Y_2\), and \(Y_3\) are order statistics, the joint probability density function for these \(Y_i\)'s obtained from a sample of 3 observations from \(f(x) = 2x, 0 < x < 1\) is given by \(f(y_1, y_2, y_3) = 6(1 - y_3)^2\) for \(0 < y_1 < y_2 < y_3 < 1\), zero elsewhere.
04

Part (b) 鈥� Step 2: Finding Marginal Distributions

The marginal distributions of \(Y_2\) and \(Y_3\) are computed by integrating the joint density over the remaining variable, resulting in \(f_{Y2}(y) = 6y(1-y)\) and \(f_{Y3}(y) = 6(1-y)^2\).
05

Part (b) 鈥� Step 3: Computing Covariance and Correlation

First compute the covariance, \(COV(Y_2, Y_3) = E(Y_2Y_3) - E(Y_2)E(Y_3)\), where the expectations are computed using the joint density. Next, using the formula for correlation, \(Corr(Y_2, Y_3) = COV(Y_2,Y_3) / (\sqrt{Var(Y_2)} \sqrt{Var(Y_3)})\), where the variances are computed from the marginal distributions. The correlation is the resulting value of this calculation.

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Most popular questions from this chapter

Let \(X\) and \(Y\) denote independent random variables with respective probability density functions \(f(x)=2 x, 0

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