4. (3 pts) Two points are picked independently and uniformly from the region inside a unit circle...
2. We distribute n points uniformly and independently on the circumference of a circle, and want to compute the probability that there is a semicircle that contain all of them. (In other words, the probability that there is a line through the center of the circle such that all n points lie on the same side of this line.) Let E be the event that such a semicircle exists. Denote by Pi, P2, ..., Pn the random points, and by...
3. In a Monte Carlo method to estimate T, we draw n points uniformly on the unit square [0, 1]2 and count how many points X fall inside the unit circle. We then multiply this number by 4 and divide by n to find an estimator of T (a) What is the probability distribution of X? b) What is the approximate distribution of 4X/n for large n? (c) For n- 1000, suppose we observed 756 points inside the unit circle....
From a given triangle of unit area, we choose two points independently with uniform distribution. The straight line connecting these points divides the triangle, with probability one, into a triangle and a quadrilateral. Calculate the expected values of the areas of these two regions.
Consider a family with eight children. Assuming that the sex each child is determined independently of the others and that each child is equally likely to be female or male, a. What is the probability that exactly four children are female? Hint: Use the binomial distribution. (10 points) b. What is the probability that at most seven children are female? Hint: What is the complement of this event? (10 points)c. Find the conditional probability that the first two children born are female...
7. (15 points) Let Xi and X2 be the position of two points drawn uniformly randomly and independently from the interval [0, 1]. Define Y = max(X,Xy) and Z-X1 + X2. (1) Calculate the joint PDF of Y and Z. (2) Derive the marginal PDF of both Y and Z. Are Y and Z independent?
7. (15 points) Let Xi and X2 be the position of two points drawn uniformly randomly and independently from the interval [0, 1]. Define Y...
Question 7. (Graded, 15 points) Randomly (i.e. uniformly) pick two points Xi and X2 from the interval (0,1), and let Y-Xi - X2] denote their distance. Calculate the PDF of Y Hint: first calculate the CDF of Z-X1 -X using any method that is proper, then find the CDF of Y 12. Finally take the derivative to find Y's PDF
28. An electronic system is composed of three components (o1. 92, and o), each of which operates independently of the other two. The electronic system has two paths available from node A to node B. C1 C2 Information is thus able to be transmitted from A to B, provided that at least one path is in operation (either 1 is operative or both o2 and c3 are in operation simultaneously). Given: P(o is in operation)0.9 P(c2 and c3 are in...
Problem 5 . This question considers uniform random points on the unit disc x2+92 〈 1 (a) A point (X, Y) is uniformly chosen in the unit disc. Find the CDF and PDF of its distance from the origin R X2 +Y2 (b) Compute the expected distance from the origin. (c) Determine the marginal PDF of X and Y (d) Are X and Y independent? (Justify your claims) e) One way to generate uniform random points on this disc is...
4. Two numbers, and y, are selected at random from the unit interval [0, 1. Find the probability that each of the three line segements so formed have length > . For example, if 0.3 and y0.6, the three line segements are (0,0.3), (0.3,0.6) and (0.6, 1.0). All three of these line segments have length >. HINT: you can solve this problem geometrically. Draw a graph of and y, with x on the horizontal axis and y on the vertical...