Chapter 3: Problem 8
Evaluate \(\int_{2}^{3} \exp \left[-2(x-3)^{2}\right] d x\)
Chapter 3: Problem 8
Evaluate \(\int_{2}^{3} \exp \left[-2(x-3)^{2}\right] d x\)
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Get started for free. Let \(X\) and \(Y\) have a bivariate normal distribution with respective
parameters \(\mu_{x}=2.8, \mu_{y}=110, \sigma_{x}^{2}=0.16,
\sigma_{y}^{2}=100\), and \(\rho=0.6 .\) Compute:
(a) \(P(106
Let the random variable \(X\) have a distribution that is \(N\left(\mu, \sigma^{2}\right)\). (a) Does the random variable \(Y=X^{2}\) also have a normal distribution? (b) Would the random variable \(Y=a X+b, a\) and \(b\) nonzero constants have a normal distribution? Hint: \(\quad\) In each case, first determine \(P(Y \leq y)\).
Let the mutually independent random variables \(X_{1}, X_{2}\), and \(X_{3}\) be \(N(0,1)\), \(N(2,4)\), and \(N(-1,1)\), respectively. Compute the probability that exactly two of these three variables are less than zero.
Suppose \(\mathbf{X}\) has a multivariate normal distribution with mean \(\mathbf{0}\) and covariance matrix $$ \boldsymbol{\Sigma}=\left[\begin{array}{llll} 283 & 215 & 277 & 208 \\ 215 & 213 & 217 & 153 \\ 277 & 217 & 336 & 236 \\ 208 & 153 & 236 & 194 \end{array}\right] $$v(a) Find the total variation of \(\mathbf{X}\) (b) Find the principal component vector Y. (c) Show that the first principal component accounts for \(90 \%\) of the total variation. (d) Show that the first principal component \(Y_{1}\) is essentially a rescaled \(\bar{X}\). Determine the variance of \((1 / 2) \bar{X}\) and compare it to that of \(Y_{1}\).
Let \(X\) and \(Y\) have a bivariate normal distribution with parameters
\(\mu_{1}=\) 3, \(\mu_{2}=1, \sigma_{1}^{2}=16, \sigma_{2}^{2}=25\), and
\(\rho=\frac{3}{5} .\) Determine the following probabilities:
(a) \(P(3
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