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Chapter 5: Problem 2
Let \(\bar{X}\) be the mean of a random sample of size \(n\) from a distribution that is \(N(\mu, 9)\). Find \(n\) such that \(P(\bar{X}-1<\mu<\bar{X}+1)=0.90\), approximately.
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Find the probability that the range of a random sample of size 4 from the
uniform distribution having the pdf \(f(x)=1,0
Let \(Y_{2}\) and \(Y_{n-1}\) denote the second and the \((n-1)\) st order statistics of a random sample of size \(n\) from a distribution of the continuous type having a distribution function \(F(x)\). Compute \(P\left[F\left(Y_{n-1}\right)-F\left(Y_{2}\right) \geq p\right]\), where \(0
Consider the following algorithm: (1) Generate \(U\) and \(V\) independent uniform \((-1,1)\) random variables. (2) Set \(W=U^{2}+V^{2}\). (3) If \(W>1\) goto Step (1). (4) Set \(Z=\sqrt{(-2 \log W) / W}\) and let \(X_{1}=U Z\) and \(X_{2}=V Z\).
Two numbers are selected at random from the interval \((0,1)\). If these values are uniformly and independently distributed, by cutting the interval at these numbers compute the probability that the three resulting line segments form a triangle.
Let two independent random samples, each of size 10, from two normal distributions \(N\left(\mu_{1}, \sigma^{2}\right)\) and \(N\left(\mu_{2}, \sigma^{2}\right)\) yield \(\bar{x}=4.8, s_{1}^{2}=8.64, \bar{y}=5.6, s_{2}^{2}=7.88\). Find a 95 percent confidence interval for \(\mu_{1}-\mu_{2}\).
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