Chapter 3

Let \(X\) have a Poisson distribution with \(\mu=100\). Use Chebyshev's inequality to determine a lower bound for \(P(75

Consider the mixture distribution \((9 / 10) N(0,1)+(1 / 10) N(0,9) .\) Show that its kurtosis is \(8.34\).

Over the years, the percentage of candidates passing an entrance exam to a prestigious law school is \(20 \%\). At one of the testing centers, a group of 50 candidates take the exam and 20 pass. Is this odd? Answer on the basis that \(X \geq 20\) where \(X\) is the number that pass in a group of 50 when the probability of a pass is \(0.2\).

If \(X\) is \(N\left(\mu, \sigma^{2}\right)\), show that \(E(|X-\mu|)=\sigma \sqrt{2 / \pi}\).

Let \(X_{1}, X_{2}\), and \(X_{3}\) be iid random variables, each with pdf \(f(x)=e^{-x}\), \(0y)=1-P\left(X_{i}>y, i=1,2,3\right)\). (b) Find the distribution of \(Y=\operatorname{maximum}\left(X_{1}, X_{2}, X_{3}\right)\).

Let \(U\) and \(V\) be independent random variables, each having a standard normal distribution. Show that the mgf \(E\left(e^{t(U V)}\right)\) of the random variable \(U V\) is \(\left(1-t^{2}\right)^{-1 / 2},-1

Let \(X\) have the conditional geometric \(\operatorname{pmf} \theta(1-\theta)^{x-1}, x=1,2, \ldots\), where \(\theta\) is a value of a random variable having a beta pdf with parameters \(\alpha\) and \(\beta\). Show that the marginal (unconditional) pmf of \(X\) is $$ \frac{\Gamma(\alpha+\beta) \Gamma(\alpha+1) \Gamma(\beta+x-1)}{\Gamma(\alpha) \Gamma(\beta) \Gamma(\alpha+\beta+x)}, \quad x=1,2, \ldots $$ If \(\alpha=1\), we obtain $$ \frac{\beta}{(\beta+x)(\beta+x-1)}, \quad x=1,2, \ldots $$ which is one form of Zipf's law.

Let \(Y\) be the number of successes throughout \(n\) independent repetitions of a random experiment with probability of success \(p=\frac{1}{4}\). Determine the smallest value of \(n\) so that \(P(1 \leq Y) \geq 0.70\)

n expression (3.4.13), the normal location model was presented. Often real data, though, have more outliers than the normal distribution allows. Based on Exercise 3.6.5, outliers are more probable for \(t\) -distributions with small degrees of freedom. Consider a location model of the form $$ X=\mu+e $$ where \(e\) has a \(t\) -distribution with 3 degrees of freedom. Determine the standard deviation \(\sigma\) of \(X\) and then find \(P(|X-\mu| \geq \sigma)\).

Let \(X\) have a gamma distribution with pdf $$ f(x)=\frac{1}{\beta^{2}} x e^{-x / \beta}, \quad 0