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Chapter 5: Problem 9

Let \(Y_{1}

Short Answer

Expert verified
The \(k\)th order statistic \(Y_{k}\) from a random sample of size \(n\) from a uniform distribution (0,1) follows a Beta distribution with parameters \(\alpha = k\) and \(\beta = n - k + 1\).

Step by step solution

01

Probabilistic representation

Let's suppose we have a random Variables \(Y_1, Y_2, ..., Y_n\) such that \(Y_1<Y_2<...<Y_n\) are uniform on the interval (0,1). Let's consider \(Y_{k}\), the \(k\)th smallest of these variables, and calculate its probability density function (pdf). We can express this as \(P(Y_k \le y) = P(\text{at least } k \text{ of the } Y_i's \text{ are less than or equal to } y)\). Because the \(Y_i\)s are independent and identically distributed, we use combinatorics and the binomial distribution to express the probability.
02

Use of combinatorial calculation

The probability that exactly \(j\) of the \(n\) variables are less than or equal to \(y\) (where \(y\) is between 0 and 1), is given by the binomial distribution with success probability \(y\), and can be expressed as \({n \choose j}y^j(1-y)^{n-j}\). Summing over \(j\) from \(k\) to \(n\) yields \(P(Y_k \le y) = \sum_{j=k}^{n} {n \choose j} y^j (1-y)^{n-j}\). This is a cumulative distribution function (CDF).
03

Deriving pdf from cdf

To get the pdf from the cdf, we take the derivative of \(P(Y_k \le y)\) with respect to \(y\). Differentiating the sum term by term, we get \(f(y) = {n \choose k}k y^{k-1}(1-y)^{n-k}\).
04

Comparison with Beta distribution

By comparison, this function \(f(y)\) is the pdf of a Beta distribution with parameters \(\alpha = k\) and \(\beta = n - k + 1\) because the Beta distribution has pdf \(f(x; \alpha, \beta) = C x^{\alpha - 1} (1 - x)^{\beta - 1}\), where \(C\) is the normalizing constant. Thus the pdf for the \(k\)th order statistic \(Y_{k}\) is a Beta(\(\alpha=k, \beta=n-k+1\)) distribution.

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Most popular questions from this chapter

Consider the sample of data: \(\begin{array}{rrrrrrrrrrr}13 & 5 & 202 & 15 & 99 & 4 & 67 & 83 & 36 & 11 & 301 \\ 23 & 213 & 40 & 66 & 106 & 78 & 69 & 166 & 84 & 64 & \end{array}\) (a) Obtain the five number summary of these data. (b) Determine if there are any outliers. (c) Boxplot the data. Comment on the plot. (d) Obtain a \(92 \%\) confidence interval for the median \(\xi_{1 / 2}\).

Let \(Y_{1}0\), provided that \(x \geq 0\), and \(f(x)=0\) elsewhere. Show that the independence of \(Z_{1}=Y_{1}\) and \(Z_{2}=Y_{2}-Y_{1}\) characterizes the gamma pdf \(f(x)\), which has parameters \(\alpha=1\) and \(\beta>0\) Hint: Use the change-of-variable technique to find the joint pdf of \(Z_{1}\) and \(Z_{2}\) from that of \(Y_{1}\) and \(Y_{2} .\) Accept the fact that the functional equation \(h(0) h(x+y) \equiv\) \(h(x) h(y)\) has the solution \(h(x)=c_{1} e^{c_{2} x}\), where \(c_{1}\) and \(c_{2}\) are constants.

Let \(X_{1}, X_{2}, \ldots, X_{n}, X_{n+1}\) be a random sample of size \(n+1, n>1\), from a distribution that is \(N\left(\mu, \sigma^{2}\right) .\) Let \(\bar{X}=\sum_{1}^{n} X_{i} / n\) and \(S^{2}=\sum_{1}^{n}\left(X_{i}-\bar{X}\right)^{2} /(n-1)\). Find the constant \(c\) so that the statistic \(c\left(\bar{X}-X_{n+1}\right) / S\) has a \(t\) -distribution. If \(n=8\), determine \(k\) such that \(P\left(\bar{X}-k S

Let \(X\) have a Poisson distribution with mean \(\theta\). Consider the simple hypothesis \(H_{0}: \theta=\frac{1}{2}\) and the alternative composite hypothesis \(H_{1}: \theta<\frac{1}{2} .\) Thus \(\Omega=\left\\{\theta: 0<\theta \leq \frac{1}{2}\right\\}\). Let \(X_{1}, \ldots, X_{12}\) denote a random sample of size 12 from this distribution. We reject \(H_{0}\) if and only if the observed value of \(Y=X_{1}+\cdots+X_{12} \leq 2\) If \(\gamma(\theta)\) is the power function of the test, find the powers \(\gamma\left(\frac{1}{2}\right), \gamma\left(\frac{1}{3}\right), \gamma\left(\frac{1}{4}\right), \gamma\left(\frac{1}{6}\right)\), and \(\gamma\left(\frac{1}{12}\right) .\) Sketch the graph of \(\gamma(\theta) .\) What is the significance level of the test?

Assume that the weight of cereal in a "10-ounce box" is \(N\left(\mu, \sigma^{2}\right)\). To test \(H_{0}: \mu=10.1\) against \(H_{1}: \mu>10.1\), we take a random sample of size \(n=16\) and observe that \(\bar{x}=10.4\) and \(s=0.4\). (a) Do we accept or reject \(H_{0}\) at the 5 percent significance level? (b) What is the approximate \(p\) -value of this test?
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