91Ó°ÊÓ

Suppose that \(F(x)\) is a cumulative distribution function. Show that (a) \(F^{\prime \prime}(x)\) and (b) \(1-[1-F(x)]^{n}\) are also cumulative distribution functions when \(n\) is a positive integer. HINT: Let \(X_{1}, \ldots, X_{n}\) be independent random variables having the common distribution function \(\vec{F}\). Define random variables \(Y\) and \(Z\) in terms of the \(X\) so that \(P\\{Y \leq x\\}=F^{n}(x)\), and \(P\\{Z \leq x\\}=1-[1-F(x)]^{n}\).

Suppose that \(10^{6}\) people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over \(\left(0,10^{6}\right)\). Let \(N\) denote the number that arrive in the first hour. Find an approximation for \(P\\{N=i\\}\)

Suppose that \(10^{6}\) people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over \(\left(0,10^{6}\right)\). Let \(N\) denote the number that arrive in the first hour. Find an approximation for \(P\\{N=i\\}\)

If \(X\) is uniformly distributed over \((0,1)\) and \(Y\) is exponentially distributed with parameter \(\dot{\lambda}=1\), find the distribution of (a) \(Z=X+Y\) and (b) \(Z=\) \(X / Y\). Assume independence.

If \(X_{1}\) and \(X_{2}\) are independent exponential random variables with respective parameters \(\bar{\lambda}_{1}\) and \(\lambda_{2}\), find the disfribution of \(Z=X_{1} / X_{2} .\) Also compute \(P\left\\{X_{1}

If \(X_{1}\) and \(X_{2}\) are independent exponential random variables with respective parameters \(\bar{\lambda}_{1}\) and \(\lambda_{2}\), find the disfribution of \(Z=X_{1} / X_{2} .\) Also compute \(P\left\\{X_{1}

Let \(X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}\) be the ordered values of \(n\) independent uniform \((0,1)\) random variables. Prove that for \(1 \leq k \leq n+1\), $$ P\left\\{X_{(k)}-X_{(k-1)}>t\right\\}=(1-t)^{n} $$ where \(X_{0} \equiv 0, X_{n+1} \equiv t\).

The expected number of typographical errors on a page of a certain magazine is \(.2\). What is the probability that an article of 10 pages contains (a) 0 , and (b) 2 or more typographical errors? Explain your reasoning!

Let \(X_{(1)} \leq X_{(2)} \leq \cdots \leq X_{(n)}\) be the ordered values of \(n\) independent uniform \((0,1)\) random variables. Prove that for \(1 \leq k \leq n+1\), $$ P\left\\{X_{(k)}-X_{(k-1)}>t\right\\}=(1-t)^{n} $$ where \(X_{0} \equiv 0, X_{n+1} \equiv t\).

The monthly worldwide average number of airplane crashes of commercial airlines is \(2.2\). What is the probability that there will be (a) more than 2 such accidents in the next month; (b) more than 4 such accidents in the next 2 months; (c) more than 5 such accidents in the next 3 months? Explain your reasoning!