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Category Archives: Algebra of events
Outcome space
Let \(\Omega\) be an outcome space with 16 outcomes. \(A\) and \(B\) are events inside of \(\Omega\). Event \(A\) has 10 outcomes and event \(B\) has 10 outcomes.
- Determine all the possible values of \(\# (A\cap B).\)
- Determine all the possible values of \(\# (A\cup B).\)
- Determine all the possible values of \(\#(A^c\cup B^c).\)
- Determine all the possible values of \(\# (A^c\cap B^c).\)
Algebras and Conditioning
Consider a deck with three cards number 1, 2, and 3. Furthermore, assume that 1 and 2 cards are colored red and the 3 card is colored black. Two of the cards are drawn with out replacement. Let \(D_1\) be the first card drawn and \(D_2\) be the second card drawn. Let \(T\) be the sum of the two cards drawn and let \(N\) be the number of red cards drawn.
- Write down the algebra of all possible event on this probability space.
- What is the algebra of events generated by \(T\), which we denote will \(\mathcal{A}(T)\) ?
- What is the algebra of events generated by \(N\) , which we denote will \(\mathcal{A}(N)\) ?
- Is \(T\) adapted to \(\mathcal{A}(N)\) ? explain in terms of the above algebras.
- Is \(N\) adapted to \(\mathcal{A}(T)\) ? explain in terms of the above algebras.
- What is \[ \mathbf{E} [ N \,|\, \mathcal{A}(T)] ? \]
- What is \[ \mathbf{E} [ T \,|\, \mathcal{A}(N)] ? \]
Algebras and Conditioning
Consider two draws from a box with replacement contain 1 red ball and 3 blue balls. Let \(X\) be number of red balls. Let \(Y\) be 1 if the two balls are the same color and 0 otherwise. Let \(Z_i\) be the random variable which returns 1 if the \(i\)-th ball is red.
- What is the sample space.
- Write down the algebra of all events on this sample space.
- What is the algebra of events generated by \(X\) ?
- What is the algebra of events generated by \(Y\) ?
- What is the algebra of events generated by \(Z_1\) ?
- What is the algebra of events generated by \(Z_2\) ?
- Which random variables are determined by an another of the random variables. Why ? How is this reflected in the algebras ?
- (*) What pair of random variables are independent ? How is this reflected in the algebras ?
Counting and geometry
1) How many ways can one order \(n\) people on a line ?
2) How many ways can one order \(n\) people on a circle or round table (note that rotations are considered equivalent )?
3) How many ways can one order \(n\) people on a circle or round table with invariance with respect to rotations and direction (clockwise versus counter-clockwise) ?
Algebra of events
Consider the sample space created by two coin flips. We will denote the outcome of first a head and then a tail by \((H,T)\). The even that there is exactly one head will be written \(\{(H,T), (T,H)\}\). Let \(\mathcal{A}\) denote the set of all events on this sample space.
- Write down all of the events in \(\mathcal{A}\). How many are there ?
- Pick a few elements of \(\mathcal{A}\) and show that the rules of being an algebra are in fact satisfied. Namely, if \(A,B \in \mathcal{A}\) then \( A \cup B \in \mathcal{A}\) and \(A^c \in \mathcal{A}\). (Here \(A^c\) is the compliment of \(A\) which is simply every event which is not is \(A\).)
- Why did I not ask you to verify by hand this property for all of the sets in \(\mathcal{A}\) ? Hint: can you tell me a lower bound on the number of such conditions you would have to check ?
- (*) Using the definition of an algebra, show that the empty set \(\emptyset\) is in \(\mathcal{A}\).
- (*) Using the definition of an algebra, show that if \(A, B \in\mathcal{A}\) then so must be \(A \cap B\).
- (*) Argue that if \(A_1, A_2,\dots,A_k \in\mathcal{A}\) then so are
\[ A_1\cup A_2\cup\cdots\cup A_k=\bigcup_{j=1}^k A_j\]
\[ A_1\cap A_2\cap\cdots\cap A_k=\bigcap_{j=1}^k A_j\]
