![4. Consider a inap φ : I 1,11 > 10, 1] defined by φ(z) :-12. Let X and Y be random variables related by the map φ, i.c., Y-o(X) (their sample spaces are then given by SX-1 1,11 and SY-10,1]). Let FY be the σ-algebra and Hy the probability measure you worked out in problem 3. Compute the adaptod ơ algebra X and the corresponding probability measure x (i.e., use the formula X (ф ія, )-, Y (S.) for any S. c ry).](http://img.homeworklib.com/questions/36a77520-785c-11ea-a1f6-cb46f42e7073.png?x-oss-process=image/resize,w_560)
![3. Let Y be a random variable and Sy 0,1] (the closed unit interval) be its sample space. ·What is the σ-algebra FY generated by the open intervals {(0, 1/2), (1/2, 1)? . If we define a probability measure y by only specifying its value on the above two open intervals (i.e., the two generators), say then can you complete the definition of : Fy 0,1] by extending it to the entire-algebra Fy? Is the extension unique?](http://img.homeworklib.com/questions/370e53f0-785c-11ea-a30e-3b3507d733f5.png?x-oss-process=image/resize,w_560)
Homework Answers
3. Let Y be a random variable and Sy 0,1] (the closed unit interval) be its...
3. Let Y be a random variable and Sy 0,1] (the closed unit interval) be its sample space. What is the o-algebra Fy generated by the open intervals 1(0, 1/2), (1/2,1)1? . If we define a probability measure y by only specifying its value on the above two open intervals (i.c., the two generators), say uy((0, 1/2)-1/3 and My((1/2, 1))-1/3, then can you complete the definition of : Fy 0,1] by extending it to the entire a-algebra Fy? Is the...
Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variabl...
Please explain
Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variable ? 2. Compute p(X0) 3. Compute E(X) . Find the PDF fxa(u) 5. Compute V(X) (Hint: use fxa found above
Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variable ? 2. Compute p(X0) 3. Compute E(X) . Find the PDF fxa(u) 5. Compute...
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e.,...
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1 with probability 1/2. Show that SZ N(0,1) 5. Let Z N(0, 1) and X = Z2. This distribution is called chi-square with degree of freedom. Calculate P(1 < X < 4) one
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1...
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero...
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero elsewhere. a) Find the value of C that makes fx(x) a valid probability density function. b) Find the cumulative distribution function of X, Fx(x). "Hint”: To double-check your answer: should be Fx(3)=0, Fx(7)=1. 1. con (continued) Consider Y=g(x)- 20 100 X 2 + Find the support (the range of possible values) of the probability distribution of Y. d) Use part (b) and the c.d.f....
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a)...
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a) Show that V 1 - U is also a uniformly distributed random variable on the interval (0,1) (b) Show that X-In(U) is an exponential random variable and find its associated parameter (c) Let W be another random variable that is uformly distributed on (0,1). Assume that U and W are independent. Show that a probability density function of Y-U+W is y, if y E...
1. Let $(x) = 2x2 and let Y = $(x). (a) Consider the case X ~U(-1,1)....
1. Let $(x) = 2x2 and let Y = $(x). (a) Consider the case X ~U(-1,1). Obtain fy and compute E[Y] (b) Now instead assume that Y ~ U(0,1/2) and that X is a continuous random variable. Explain carefully why it is possible to choose fx such that fx (2) = 0 whenever 21 > 1. Obtain an expression linking fx(2) to fx(-x) for 3 € (-1,1). Show that E[X] = -2/3 + 2 S xfx(x) dx. Using your expression...
please answer correctly and show work 10. (12 points, 6+6) Let the joint probability density function...
please answer correctly and show work
10. (12 points, 6+6) Let the joint probability density function of a bivariate continuous random variable (X,Y) is defined by Sc(x + y) if 0 Sy su <1, f(x,y) = 3 0 otherwise. (1). Find the value of constant c. (2). Find E(X)
1) 2) 3) 4) 5) Suppose that X is a uniform random variable on the interval...
1)
2)
3)
4)
5)
Suppose that X is a uniform random variable on the interval (0, 1) and let Y = 1/X. a. Give the smallest interval in which Y is guaranteed to be. Enter -Inf or Inf for – or o. Interval:( b. Compute the probability density function of Y on this interval. fy(y) = Suppose that X ~ Bin(4, 1/3). Find the probability mass function of Y = (X – 2)2. a. List all possible values that...
(a)-(d)? Problem(11) (10 points) Let Z~Normal(0, 1). Recall the definition of -value, i.e., P(Z>)-r. (a) (1...
(a)-(d)?
Problem(11) (10 points) Let Z~Normal(0, 1). Recall the definition of -value, i.e., P(Z>)-r. (a) (1 point) Find the probability of P(-2a/2<Z < 2a/2) (b) (3 points) Let X1, X2, , Xa be a random sample from some known) mean p and (known) variance o2. Based on the Central Limit Theorm and part (a) above, show that the confidence intervals for the population mean u can be estimated by population with (un- P(x- <pAX+Za/2 =1-a. Za/2 (c) (2 points) The...
I am studying Continuous Random Variables. Hope can some one tell me the solutions of these two problems! II.1 Let X be...
I am studying Continuous Random Variables.
Hope can some one tell me the solutions of these two
problems!
II.1 Let X be a continuous random variable with the density function 1/4 if x E (-2,2) 0 otherwise &Cx)={ Find the probability density function of Z = X density function fx. Find the distribution function Fy (t) and the density function f,(t) of Y=지 (in terms of Fx and fx).
II.1 Let X be a continuous random variable with the density...
3. Let Y be a random variable and Sy 0,1] (the closed unit interval) be its sample space. What is the o-algebra Fy generated by the open intervals 1(0, 1/2), (1/2,1)1? . If we define a probability measure y by only specifying its value on the above two open intervals (i.c., the two generators), say uy((0, 1/2)-1/3 and My((1/2, 1))-1/3, then can you complete the definition of : Fy 0,1] by extending it to the entire a-algebra Fy? Is the...
Please explain
Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variable ? 2. Compute p(X0) 3. Compute E(X) . Find the PDF fxa(u) 5. Compute V(X) (Hint: use fxa found above
Let Z N(0,1), and let X = max(Z, 0) 1. Find Fx in terms of Φ(t). Ís X a continuous random variable ? 2. Compute p(X0) 3. Compute E(X) . Find the PDF fxa(u) 5. Compute...
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1 with probability 1/2. Show that SZ N(0,1) 5. Let Z N(0, 1) and X = Z2. This distribution is called chi-square with degree of freedom. Calculate P(1 < X < 4) one
N(0, 1) and let S be a 4. Let Z random sign independent of Z, i.e., S is 1 with probability 1/2 and -1...
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero elsewhere. a) Find the value of C that makes fx(x) a valid probability density function. b) Find the cumulative distribution function of X, Fx(x). "Hint”: To double-check your answer: should be Fx(3)=0, Fx(7)=1. 1. con (continued) Consider Y=g(x)- 20 100 X 2 + Find the support (the range of possible values) of the probability distribution of Y. d) Use part (b) and the c.d.f....
1. Let U be a random variable that is uniformly distributed on the interval (0,1) (a) Show that V 1 - U is also a uniformly distributed random variable on the interval (0,1) (b) Show that X-In(U) is an exponential random variable and find its associated parameter (c) Let W be another random variable that is uformly distributed on (0,1). Assume that U and W are independent. Show that a probability density function of Y-U+W is y, if y E...
1. Let $(x) = 2x2 and let Y = $(x). (a) Consider the case X ~U(-1,1). Obtain fy and compute E[Y] (b) Now instead assume that Y ~ U(0,1/2) and that X is a continuous random variable. Explain carefully why it is possible to choose fx such that fx (2) = 0 whenever 21 > 1. Obtain an expression linking fx(2) to fx(-x) for 3 € (-1,1). Show that E[X] = -2/3 + 2 S xfx(x) dx. Using your expression...
please answer correctly and show work
10. (12 points, 6+6) Let the joint probability density function of a bivariate continuous random variable (X,Y) is defined by Sc(x + y) if 0 Sy su <1, f(x,y) = 3 0 otherwise. (1). Find the value of constant c. (2). Find E(X)
1)
2)
3)
4)
5)
Suppose that X is a uniform random variable on the interval (0, 1) and let Y = 1/X. a. Give the smallest interval in which Y is guaranteed to be. Enter -Inf or Inf for – or o. Interval:( b. Compute the probability density function of Y on this interval. fy(y) = Suppose that X ~ Bin(4, 1/3). Find the probability mass function of Y = (X – 2)2. a. List all possible values that...
(a)-(d)?
Problem(11) (10 points) Let Z~Normal(0, 1). Recall the definition of -value, i.e., P(Z>)-r. (a) (1 point) Find the probability of P(-2a/2<Z < 2a/2) (b) (3 points) Let X1, X2, , Xa be a random sample from some known) mean p and (known) variance o2. Based on the Central Limit Theorm and part (a) above, show that the confidence intervals for the population mean u can be estimated by population with (un- P(x- <pAX+Za/2 =1-a. Za/2 (c) (2 points) The...
I am studying Continuous Random Variables.
Hope can some one tell me the solutions of these two
problems!
II.1 Let X be a continuous random variable with the density function 1/4 if x E (-2,2) 0 otherwise &Cx)={ Find the probability density function of Z = X density function fx. Find the distribution function Fy (t) and the density function f,(t) of Y=지 (in terms of Fx and fx).
II.1 Let X be a continuous random variable with the density...
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