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5 (10 points). We start with a stick of length L. Break it at a point...
A stick is broken into three pieces at two randomly chosen points on the stick. What is the proba...
A stick is broken into three pieces at two randomly chosen points on the stick. What is the probability that no piece is longer than half the length of the stick? To do this problem, it is useful to split it into the following steps (assuming the length of stick is 1). (a) Let U1 and U2 are independent and uniformly distributed on (0, 1). Define X = min(U1, U2) and Y = max(U1, U2). Use the fact that P(a...
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...
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...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4. 3 marks (b) Suppose X1 and X2 are two iid normal N(μ, σ*) variables. Define Are random variables V and W independent? Mathematically justify your answer 3 marks (c) Let C denote the unit circle...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the pr...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4 3 marks (b) Suppose X, and X2 are two iid normal N(μ, σ2) variables. Define Are random variables V and W independent? Mathematically justify your answer. 3 marks] (c) Let C denote the unit circle...
5. (40 points) Let f(x,y) = (x + y),0 < 2,2 <y < 1 be the...
5. (40 points) Let f(x,y) = (x + y),0 < 2,2 <y < 1 be the joint pdf of X and Y. (1) Find the marginal probability density functions fx(x) and fy(y). (2) Find the means hx and my. (3) Find P(X>01Y > 0.5). (4) Find the correlation coefficient p.
(1 point) Two points are selected randomly on a line of length 10 so as to...
(1 point) Two points are selected randomly on a line of length 10 so as to be on opposite sides of the midpoint of the line. In other words, the two points X and Y are independent random variables such that X is uniformly distributed over [0,5) and Y is uniformly distributed over (5, 10]. Find the probability that the distance between the two points is greater than 2. answer:
Question 7. (Graded, 15 points) Randomly (i.e. uniformly) pick two points Xi and X2 from the...
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
please make sure to give the answer for the second question. #2. (24 points) Let X...
please make sure to give the answer for the second
question.
#2. (24 points) Let X and Y have joint density f(x,y) = 60xy2, 0<x<1, 0<y<1, x+y<1. (a) Find the marginal pdf of Y. Use it to find E(Y) (b) Give an integral expression for P(X + Y <0.75), but do not evaluate. (C) Give an integral expression for E(XY), but do not evaluate. Optional two point bonus problem. In Problem 2 above, is the distribution of Y skewed to...
(1 point) Two points are selected randomly on a line of length 16 so as to...
(1 point) Two points are selected randomly on a line of length 16 so as to be on opposite sides of the midpoint of the line. In other words, the two points X and Y are independent random variables such that X is uniformly distriuted over [0,8) and Y is uniformly distributed over (8,16] Find the probability that the distance between the two points is greater than 6. P(|X – Y| > 6) =
We start out with a couple of defintions and examples. Definition: Let X and Y have...
We start out with a couple of defintions and examples. Definition: Let X and Y have joint pdf f(x,y). The conditional pdf of Y given X = x (resp. of X given Y = y) is defined by h(y|x) = f (x, y) resp. g(x|y) = f (x, y) f1(x) f2(y) If A is a subset of the real line, then P(Y ∈A|X =x)= h(y|x)dy resp. P(X ∈A|Y =y)= g(x|y)dx . AA Example 1 (seen in class) Consider the joint...
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...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4. 3 marks (b) Suppose X1 and X2 are two iid normal N(μ, σ*) variables. Define Are random variables V and W independent? Mathematically justify your answer 3 marks (c) Let C denote the unit circle...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4 3 marks (b) Suppose X, and X2 are two iid normal N(μ, σ2) variables. Define Are random variables V and W independent? Mathematically justify your answer. 3 marks] (c) Let C denote the unit circle...
5. (40 points) Let f(x,y) = (x + y),0 < 2,2 <y < 1 be the joint pdf of X and Y. (1) Find the marginal probability density functions fx(x) and fy(y). (2) Find the means hx and my. (3) Find P(X>01Y > 0.5). (4) Find the correlation coefficient p.
(1 point) Two points are selected randomly on a line of length 10 so as to be on opposite sides of the midpoint of the line. In other words, the two points X and Y are independent random variables such that X is uniformly distributed over [0,5) and Y is uniformly distributed over (5, 10]. Find the probability that the distance between the two points is greater than 2. answer:
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
please make sure to give the answer for the second
question.
#2. (24 points) Let X and Y have joint density f(x,y) = 60xy2, 0<x<1, 0<y<1, x+y<1. (a) Find the marginal pdf of Y. Use it to find E(Y) (b) Give an integral expression for P(X + Y <0.75), but do not evaluate. (C) Give an integral expression for E(XY), but do not evaluate. Optional two point bonus problem. In Problem 2 above, is the distribution of Y skewed to...
(1 point) Two points are selected randomly on a line of length 16 so as to be on opposite sides of the midpoint of the line. In other words, the two points X and Y are independent random variables such that X is uniformly distriuted over [0,8) and Y is uniformly distributed over (8,16] Find the probability that the distance between the two points is greater than 6. P(|X – Y| > 6) =
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