Probabilities and Set Theory

Please see the attached file for the fully formatted problems. Let (Omega, A, P) be a probability space. We consider a series of mesurable sets (An)nEnCA . Prove that P(lim infnAn).... Prove that if the series is convergent, we have continuity, i.e. ...

Proof : Lebesgue Measure

Show that if mu is the Lebesgue measure on R^2 (R = set of real numbers), then for every line D we have mu(D) = 0 .

Probabilities and Set Theory

We say that an event A E A is nearly certain if A is nearly certainly equal to OMEGA. In other words, OMEGA = AUN , where N is a negligeable set.

Probabilities : Darts on a Dartboard

My uncle plays darts on a circular dart board of radius 20 cm. He assumes a dart lands anywhere on the board with equal probability. a) What is the probability that his dart lands less than 5 cm from the centre of the board? b) That his dart lands exactly 5 cm from the centre? c) My uncle wants to divide his board into ten ...continues

Vectors : Equations of Planes and Lines.

Please see the attached file for the fully formatted problems. VECTOR PROBLEMS 3 (1) Find the equation of the plane that contains the line r = (1, -3, 4) + s (2, 1, 1) and is parallel to the line r = t (1, 2, 3). Write the equation in the form r . n = λ (2) The line l has equ ...continues

Cartesian product of finitely many countable sets

I need help constructing a proof for the Cartesian product of finitely many countable sets is countable.

Cardinality

1. Find the cardinality of the set of all irrational numbers, and prove your answer is correct. 2a. Is there a line in the x-y plane such that both coordinates of every point on the line are rational? Prove your answer is correct. 2b. Find the cardinality of the set of all complex numbers, and justify your answer. 3a. W ...continues

Accounting taxes

1. Does your employer pay taxes on your behalf? Explain your answer.

Long-term notes

2. What are examples of long-term notes payable in our personal finances?

Accounting

3) Would you rather have a hundred dollars today or a hundred dollars a year from now? Why?