Problem 1
Consider a game based on the days of a 31 -day month. A day is chosen at randomsay, by spinning a spinner. The prize is a number of dollars equal to the sum of the digits in the date of the chosen day, For example, choosing the 31 st of the month pays \(\$ 3+\$ 1=\$ 4,\) as does choosing the fourth day of the month. (a) Set up the underlying sample space \(\Omega\) and its probability density, the value of which at \(\omega\) gives the reward associated with \(\omega\). (b) Define a random variable \(X(\omega)\) on \(\Omega\) with a value at \(\omega\) that gives the reward associated with \(\omega\). (c) Set up a sample space \(\Omega_{X}\) consisting of the elements in the range of \(X,\) and give the probability distribution \(p x\) on \(\Omega_{X}\) arising from \(X\). (d) Determine \(P(X=6)\). (e) Determine \(P(2 \leq X \leq 4)=P(\omega: 2 \leq X(\omega) \leq 4)\). (f) Determine \(P(X>10)=P(\omega: X(\omega)>10)\).
Problem 1
Suppose \(S\) is a set with \(k\) elements. How many elements are in \(S^{n}\), the cross product \(S \times S \times \cdots \times S\) of \(n\) copies of \(S ?\)
Problem 1
Compute the variance \(\operatorname{Var}(X)\) of the random variable \(X\) that counts the number of heads in four flips of a fair coin.
Problem 2
Suppose that \(\sum_{l=1}^{n} a_{i}=\sum_{j=1}^{m} b_{j}=1\) where \(0 \leq a_{i}, b_{j} \leq 1 .\) Use the Product of Sums Principle to prove that \(\sum_{i=1}^{n} \sum_{j=1}^{m} a_{i} \cdot b_{j}=1 .\) Does the result hold if some of the \(a_{i}\) and \(b_{j}\) can be less than zero and greater than one?
Problem 2
Compute the variance \(\operatorname{Var}(X)\) of the random variable \(X\) that counts the number of heads in four flips of a coin that lands heads with a frequency of \(1 / 3 .\)
Problem 2
Under which of the following circumstances is the pair \(A\), \(B\) of events in sample space \(\Omega\) an independent pair? Explain your answer. (a) \(A\) and \(B\) are disjoint, \(P(A)>0,\) and \(P(B)>0\) (b) \(P(A)=0\) and \(P(B)>0\) (c) \(P(A)=P(B)=0\)
Problem 2
Suppose that sample space \(\Omega_{1}\) is chosen to model the experiment of rolling a pair of dice and that the probability density function \(p\) assigned to \(\Omega_{1}\) is \(p(\omega)=1 / 36\) for \(\omega \in \Omega_{1}\). Under these assumptions, compute the probability of rolling a sum of 3. Compare your answer to the answers of \(1 / 18\) and \(1 / 11\) obtained in the text, and discuss.
Problem 3
Suppose \(\sum_{i=1}^{n} a_{i}=2, \sum_{j=1}^{m} b_{j}=3,\) and \(\sum_{k=1}^{l} c_{k}=5 .\) Evaluate $$ \sum_{i=1}^{n} a_{i}\left(\sum_{j=1}^{m} \sum_{k=1}^{l} b_{j} \cdot c_{k}\right) $$
Problem 3
Suppose \(A\) and \(B\) are disjoint events in a sample space \(\Omega\). Is it possible that \(A\) and \(B\) could be independent? Explain your answer.
Problem 3
Define a random variable \(X\) on the sample space \(\Omega\) by setting \(X(\omega)=3\) for all \(\omega \in \Omega\). What is \(E(X) ? \operatorname{Var}(X) ?\)
