Chapter 4: Problem 5
Let \(X_{1}\) and \(X_{2}\) be two independent random variables so that the variances of \(X_{1}\) and \(X_{2}\) are \(\sigma_{1}^{2}=k\) and \(\sigma_{2}^{2}=2\), respectively. Given that the variance of \(Y=3 X_{2}-X_{1}\) is 25, find \(k\)
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Let \(\bar{X}_{n}\) denote the mean of a random sample of size \(n\) from a distribution that is \(N\left(\mu, \sigma^{2}\right) .\) Find the limiting distribution of \(\bar{X}_{n}\).
Let \(Y\) denote the sum of the observations of a random sample of size 12 from
a distribution having pmf \(p(x)=\frac{1}{6}, x=1,2,3,4,5,6\), zero elsewhere.
Compute an approximate value of \(P(36 \leq Y \leq 48)\). Hint: Since the event
of interest is \(Y=36,37, \ldots, 48\), rewrite the probability as
\(P(35.5
Let \(X\) and \(Y\) be random variables with \(\mu_{1}=1, \mu_{2}=4, \sigma_{1}^{2}=4, \sigma_{2}^{2}=\) 6, \(\rho=\frac{1}{2}\). Find the mean and variance of \(Z=3 X-2 Y\).
Let \(X_{1}, \ldots, X_{n}\) be a random sample from a uniform \((a, b)\) distribution. Let \(Y_{1}=\min X_{i}\) and let \(Y_{2}=\max X_{i} .\) Show that \(\left(Y_{1}, Y_{2}\right)^{\prime}\) converges in probability to the vector \((a, b)^{\prime}\).
Let \(Y\) be \(b\left(72, \frac{1}{3}\right)\). Approximate \(P(22 \leq Y \leq 28)\).
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