Chapter 3: Problem 7
Show that the graph of a pdf \(N\left(\mu, \sigma^{2}\right)\) has points of inflection at \(x=\mu-\sigma\) and \(x=\mu+\sigma\).
Chapter 3: Problem 7
Show that the graph of a pdf \(N\left(\mu, \sigma^{2}\right)\) has points of inflection at \(x=\mu-\sigma\) and \(x=\mu+\sigma\).
All the tools & learning materials you need for study success - in one app.
Get started for freeThe mgf of a random variable \(X\) is \(e^{4\left(e^{t}-1\right)} .\) Show that
\(P(\mu-2 \sigma
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
. Let \(X\) and \(Y\) have the joint \(\operatorname{pmf} p(x, y)=e^{-2} /[x !(y-x) !], y=0,1,2, \ldots ;\) \(x=0,1, \ldots, y\), zero elsewhere. (a) Find the mgf \(M\left(t_{1}, t_{2}\right)\) of this joint distribution. (b) Compute the means, the variances, and the correlation coefficient of \(X\) and \(Y\). (c) Determine the conditional mean \(E(X \mid y)\). Hint: Note that $$ \sum_{x=0}^{y}\left[\exp \left(t_{1} x\right)\right] y ! /[x !(y-x) !]=\left[1+\exp \left(t_{1}\right)\right]^{y} $$ Why?
For the Burr distribution, show that $$ E\left(X^{k}\right)=\frac{1}{\beta^{k / \tau}} \Gamma\left(\alpha-\frac{k}{\tau}\right) \Gamma\left(\frac{k}{\tau}+1\right) / \Gamma(\alpha) $$ provided \(k<\alpha \tau\)
Let \(X_{1}, X_{2}, X_{3}\) be iid random variables each having a standard normal distribution. Let the random variables \(Y_{1}, Y_{2}, Y_{3}\) be defined by $$ X_{1}=Y_{1} \cos Y_{2} \sin Y_{3}, \quad X_{2}=Y_{1} \sin Y_{2} \sin Y_{3}, \quad X_{3}=Y_{1} \cos Y_{3} $$ where \(0 \leq Y_{1}<\infty, 0 \leq Y_{2}<2 \pi, 0 \leq Y_{3} \leq \pi .\) Show that \(Y_{1}, Y_{2}, Y_{3}\) are mutually independent.
What do you think about this solution?
We value your feedback to improve our textbook solutions.