Chapter 3: Problem 9
Toss two nickels and three dimes at random. Make appropriate assumptions and compute the probability that there are more heads showing on the nickels than on the dimes.
Chapter 3: Problem 9
Toss two nickels and three dimes at random. Make appropriate assumptions and compute the probability that there are more heads showing on the nickels than on the dimes.
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2^{-x^{2}},-\infty
Compute the measures of skewness and kurtosis of a gamma distribution which has parameters \(\alpha\) and \(\beta\).
. Let \(X_{1}, X_{2}\), and \(X_{3}\) be three independent chi-square variables with \(r_{1}, r_{2}\), and \(r_{3}\) degrees of freedom, respectively. (a) Show that \(Y_{1}=X_{1} / X_{2}\) and \(Y_{2}=X_{1}+X_{2}\) are independent and that \(Y_{2}\) is \(\chi^{2}\left(r_{1}+r_{2}\right)\) (b) Deduce that $$ \frac{X_{1} / r_{1}}{X_{2} / r_{2}} \quad \text { and } \quad \frac{X_{3} / r_{3}}{\left(X_{1}+X_{2}\right) /\left(r_{1}+r_{2}\right)} $$ are independent \(F\) -variables.
Compute \(P\left(X_{1}+2 X_{2}-2 X_{3}>7\right)\), if \(X_{1}, X_{2}, X_{3}\) are iid with common distribution \(N(1,4)\).
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 \(Y_{1}\).
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