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Chapter 4: Problem 26

Compute \(P\left(Y_{3}<\xi_{0.5}

Short Answer

Expert verified
The probability \(P\left(Y_{3}<\xi_{0.5}<Y_{7}\right) = \frac{5}{9}\

Step by step solution

01

Assumption Validation

First, it is important to validate the condition provided that \(Y_{1}<\cdots<Y_{9}\) are the order statistics of a random sample of size 9 from a continuous distribution. This means they are ranked in increasing order from smallest to largest.
02

Understanding the Median

The median of a distribution lies at the point where half the observations are greater and half are smaller. It is often denoted with \(\xi_{0.5}\). With nine observations in numerical order, the median value falls at \(Y_5\). This means, \(Y_{3}< \xi_{0.5}< Y_{7}\) is the range within which the median value lies.
03

Computing the Probability

Since the distribution is continuous, each order statistic is equally likely to be the median. Therefore, there are 5 equally likely places for the median to be: \(Y_3\), \(Y_4\), \(Y_5\), \(Y_6\), or \(Y_7\). Thus, the probability requested is simply the sum of the probabilities for each of these cases. Since these are equally likely, each event has a probability of \(1/{9}\), for the 9 positions the median value can take in the ordered sequence. So, the total probability is \(P\left(Y_{3}<\xi_{0.5}<Y_{7}\right) = 5 \times \frac{1}{9}\)

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Most popular questions from this chapter

Let \(z^{*}\) be drawn at random from the discrete distribution which has mass \(n^{-1}\) at each point \(z_{i}=x_{i}-\bar{x}+\mu_{0}\), where \(\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) is the realization of a random sample. Determine \(E\left(z^{*}\right)\) and \(V\left(z^{*}\right)\).

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In the baseball data set discussed in the last exercise, it was found that out of the 59 baseball players, 15 were left-handed. Is this odd, since the proportion of left-handed males in America is about \(11 \% ?\) Answer by using \((4.2 .7)\) to construct a \(95 \%\) approximate confidence interval for \(p\), the proportion of left-handed baseball players.

The data set on Scottish schoolchildren discussed in Example 4.1.5 included the eye colors of the children also. The frequencies of their eye colors are \(\begin{array}{lccl}\text { Blue } & \text { Light } & \text { Medium } & \text { Dark } \\ 2978 & 6697 & 7511 & 5175\end{array}\)

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