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Chapter 4: Problem 17

Let \(Y_{1}

Short Answer

Expert verified
This problem requires understanding the concept of joint and conditional pdf and the properties of the given distribution. For (a) the solution involves calculating the joint pdf of \(Y_{3}\) and \(Y_{4}\) given the expression for the pdf of the order statistics and using the given pdf and its cdf. For (b) the solution involves finding the marginal pdf for \(Y_{4}\) and calculating the ratio of joint pdf and marginal pdf to get the conditional pdf of \(Y_{3}\) given \(Y_{4} = y_4\). Lastly, for (c) the solution requires calculating the expected value (or average) of \(Y_{3}\) given \(Y_{4}=y_{4}\), based on the conditional pdf from (b).

Step by step solution

01

- Find the joint pdf of \(Y_{3}\) and \(Y_{4}\)

To find the joint pdf of \(Y_{3}\) and \(Y_{4}\), we apply the definition of the joint pdf for order statistics, which is given by: \(f_{Y3, Y4}(y3, y4) = \frac{n!}{(k-1)!(n-k)!(j-1)!(k-j)!(n-m)!(m-j)!} * [F(y_j) - F(y_{j-1})]^{j-1} * [F(y_k) - F(y_j)]^{k-j} * [1 - F(y_k)]^{n-k} * f(y_j) * f(y_k)\). Here, \(f(x) = 2x\) for \(0 < x < 1\) is the pdf and \(F(x) = x^2\) for \(0 < x < 1\) is the corresponding cdf. We substitute \(n=4\), \(k=4\), \(j=3\) into the formula and calculate the values for \(f_{Y3, Y4}(y3, y4)\).
02

- Find the Conditional pdf of \(Y_{3}\) given \(Y_{4}=y_{4}\)

For the conditional pdf we use the result from step 1 and divide the joint pdf by the marginal pdf of \(Y_{4}\), which can be obtained using its definition. Thus, \(f_{Y3| Y4}(y3|y4) = \frac{f_{Y3, Y4}(y3, y4)}{f_{Y4}(y4)}\) . After calculating the marginal pdf and dividing the joint pdf by the marginal pdf, we get the conditional pdf.
03

- Evaluate the conditional expectation \(E\left(Y_{3} \mid y_{4}\right)\)

The conditional expectation \(E(Y_{3}\mid Y_{4}=y_{4})\) is the average value that \(Y_{3}\) will take on given that \(Y_{4}\) has taken on the value \(y_{4}\). This can be computed by taking the integral over the range of \(Y_{3}\): \(E(Y_{3}|Y_{4}=y_{4}) = \int_{-\infty}^{\infty} y3*f_{Y3| Y4}(y3|y4)dy3\). Using the conditional pdf we found in step 2, we calculate the integral (note that the range of \(Y_3\) is from 0 to \(y_4\), given \(Y_4 = y_4\)).

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Most popular questions from this chapter

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