Problem 8
A bowl contains 10 chips, of which 8 are marked $$\$ 2$$ each and 2 are marked $$\$ 5$$ each. Let a person choose, at random and without replacement, three chips from this bowl. If the person is to receive the sum of the resulting amounts, find his expectation.
Problem 8
Let \(X\) be a random variable such that \(E\left[(X-b)^{2}\right]\) exists for all real \(b\). Show that \(E\left[(X-b)^{2}\right]\) is a minimum when \(b=E(X)\).
Problem 9
Let \(f(x)=2 x, 0
Problem 9
Let \(X\) have the \(\operatorname{pmf} p(x)=1 / 3, x=-1,0,1\). Find the pmf of \(Y=X^{2}\).
Problem 9
Let \(X\) be a random variable of the continuous type that has pdf \(f(x)\). If \(m\) is the unique median of the distribution of \(X\) and \(b\) is a real constant, show that $$ E(|X-b|)=E(|X-m|)+2 \int_{m}^{b}(b-x) f(x) d x $$ provided that the expectations exist. For what value of \(b\) is \(E(|X-b|)\) a minimum?
Problem 9
Bowl I contains six red chips and four blue chips. Five of these 10 chips are selected at random and without replacement and put in bowl II, which was originally empty. One chip is then drawn at random from bowl II. Given that this chip is blue, find the conditional probability that two red chips and three blue chips are transferred from bowl I to bowl II.
Problem 9
The median and quantiles, in general, are discussed in Section 1.7.1. Find the
median of each of the following distributions:
(a) \(p(x)=\frac{4 !}{x !(4-x)
!}\left(\frac{1}{4}\right)^{x}\left(\frac{3}{4}\right)^{4-x}, x=0,1,2,3,4\),
zero elsewhere.
(b) \(f(x)=3 x^{2}, 0
Problem 9
For every one-dimensional set \(C\) for which the integral exists, let \(Q(C)=\)
\(\int_{C} f(x) d x\), where \(f(x)=6 x(1-x), 0
Problem 9
Consider an urn that contains slips of paper each with one of the numbers \(1,2, \ldots, 100\) on it. Suppose there are \(i\) slips with the number \(i\) on it for \(i=1,2, \ldots, 100\). For example, there are 25 slips of paper with the number \(25 .\) Assume that the slips are identical except for the numbers. Suppose one slip is drawn at random. Let \(X\) be the number on the slip. (a) Show that \(X\) has the \(\operatorname{pmf} p(x)=x / 5050, x=1,2,3, \ldots, 100\), zero elsewhere. (b) Compute \(P(X \leq 50)\). (c) Show that the cdf of \(X\) is \(F(x)=[x]([x]+1) / 10100\), for \(1 \leq x \leq 100\), where \([x]\) is the greatest integer in \(x\).
Problem 10
Let \(X\) have the pmf $$ p(x)=\left(\frac{1}{2}\right)^{|x|}, \quad x=-1,-2,-3, \ldots $$ Find the pmf of \(Y=X^{4}\).
