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bc Let (X, Y) denote the numbers of goals scored by teams A and B respectively...
PROBLEM 2 Two teams A and B play a soccer match. The number of goals scored...
PROBLEM 2 Two teams A and B play a soccer match. The number of goals scored by Team A is modeled by a Poisson process Ni(t) with rate l1 = 0.02 goals per minute, and the number of goals scored by Team B is modeled by a Poisson process N2(t) with rate 12 = 0.03 goals per minute. The two processes are assumed to be independent. Let N(t) be the total number of goals in the game up to and...
Let X and Y denote independent random variables with respective probability density functions, f(x) = 2x,...
Let X and Y denote independent random variables with respective probability density functions, f(x) = 2x, 0<x<1 (zero otherwise), and g(y) = 3y2, 0<y<1 (zero otherwise). Let U = min(X,Y), and V = max(X,Y). Find the joint pdf of U and V.
independence Ex: 46 Let X, Y be independent Poisson r.v. with parameters x,, ta respectively. Compute...
independence Ex: 46 Let X, Y be independent Poisson r.v. with parameters x,, ta respectively. Compute P (2X=k 1X+ = P({X= k} n{X+Y=n} PL&X=k} ^{ Y = n-k}) v 3 PL&X+Y= n3) Pl{X+Y=n}) (EV-n-ks) 1 to. Ank. Kle ni la ik' (n-kel! Continen) Gent This is Binth, t)! n - V tylne - (), the) HMW: By using the interpretation of Poisson & Binomial random variables, could we have guessed this result!
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X),...
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X), a and b are constants such that 1 S a <b. Find the confidence level of this interval. Xi, X, want to test H0: θ-ya versus H1: θ> %. Suppose we set our decision rule as reject Ho , X, be a random sample from the Uniform (0, θ) distribution. Consider (b) ,X5 is a random sample from the Bernoulli (0) distribution, 0 <...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
10 marks Let X~ Poisson(A), which has density 5 marks Find the relative errors when P(O Y 3 2) is approximated by using the standard normal distribution, for λ = 1, 102, 104, 106, respectively (For...
10 marks Let X~ Poisson(A), which has density 5 marks Find the relative errors when P(O Y 3 2) is approximated by using the standard normal distribution, for λ = 1, 102, 104, 106, respectively (For u 0, the relative error is defined as E-11-aa statistical software to find probability values.) Vapprox/v. You may use R or any
10 marks Let X~ Poisson(A), which has density
5 marks Find the relative errors when P(O Y 3 2) is approximated by...
Consider the following linear regression model 1. For any X = x, let Y = xB+U,...
Consider the following linear regression model 1. For any X = x, let Y = xB+U, where B erk. 2. X is exogenous. 3. The probability model is {f(u; ) is a distribution on R: Ef [U] = 0, VAR; [U] = 62,0 >0}. 4. Sampling model: {Y}}}=1 is an independent sample, sequentially generated using Y; = xiß +Ui, where the U; are IID(0,62). (i) Let K > 0 be a given number. We wish to estimate B using least-squares...
Question 1(a&b) Question 3 (a,b,c,d) QUESTION 1 (15 MARKS) Let X and Y be continuous random variables with joint probability density function 6e.de +3,, х, у z 0 otherwise f(x, y 0 Det...
Question 1(a&b)
Question 3 (a,b,c,d)
QUESTION 1 (15 MARKS) Let X and Y be continuous random variables with joint probability density function 6e.de +3,, х, у z 0 otherwise f(x, y 0 Determine whether or not X and Y are independent. (9 marks) a) b) Find P(x> Y). Show how you get the limits for X and Y (6 marks) QUESTION 3 (19 MARKS) Let f(x, x.) = 2x, , o x, sk: O a) Find k xsl and f(x,...
Question 1 V g(x) O a Let g)-IOdt, where asx Sb. The figure above shows the graph of g on [a.t]. Which of the following could be the graph of f on [a,b]? (B) y (A) Y 0 ct (E) Y D) Question 2 100 12 1...
Question 1 V g(x) O a Let g)-IOdt, where asx Sb. The figure above shows the graph of g on [a.t]. Which of the following could be the graph of f on [a,b]? (B) y (A) Y 0 ct (E) Y D) Question 2 100 12 18 24 Hours The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown above. Of the following, which best approximates the total number of...
PROBLEM 2 Two teams A and B play a soccer match. The number of goals scored by Team A is modeled by a Poisson process Ni(t) with rate l1 = 0.02 goals per minute, and the number of goals scored by Team B is modeled by a Poisson process N2(t) with rate 12 = 0.03 goals per minute. The two processes are assumed to be independent. Let N(t) be the total number of goals in the game up to and...
independence Ex: 46 Let X, Y be independent Poisson r.v. with parameters x,, ta respectively. Compute P (2X=k 1X+ = P({X= k} n{X+Y=n} PL&X=k} ^{ Y = n-k}) v 3 PL&X+Y= n3) Pl{X+Y=n}) (EV-n-ks) 1 to. Ank. Kle ni la ik' (n-kel! Continen) Gent This is Binth, t)! n - V tylne - (), the) HMW: By using the interpretation of Poisson & Binomial random variables, could we have guessed this result!
1(a) Let Xi, X2, the random interval (ay,, b%) around 9, where Y, = max(Xi,X2 ,X), a and b are constants such that 1 S a <b. Find the confidence level of this interval. Xi, X, want to test H0: θ-ya versus H1: θ> %. Suppose we set our decision rule as reject Ho , X, be a random sample from the Uniform (0, θ) distribution. Consider (b) ,X5 is a random sample from the Bernoulli (0) distribution, 0 <...
Return to the original model. We now introduce a Poisson intensity parameter X for every time point and denote the parameter () that gives the canonical exponential family representation as above by θ, . We choose to employ a linear model connecting the time points t with the canonical parameter of the Poisson distribution above, i.e., n other words, we choose a generalized linear model with Poisson distribution and its canonical link function. That also means that conditioned on t,...
10 marks Let X~ Poisson(A), which has density 5 marks Find the relative errors when P(O Y 3 2) is approximated by using the standard normal distribution, for λ = 1, 102, 104, 106, respectively (For u 0, the relative error is defined as E-11-aa statistical software to find probability values.) Vapprox/v. You may use R or any
10 marks Let X~ Poisson(A), which has density
5 marks Find the relative errors when P(O Y 3 2) is approximated by...
Consider the following linear regression model 1. For any X = x, let Y = xB+U, where B erk. 2. X is exogenous. 3. The probability model is {f(u; ) is a distribution on R: Ef [U] = 0, VAR; [U] = 62,0 >0}. 4. Sampling model: {Y}}}=1 is an independent sample, sequentially generated using Y; = xiß +Ui, where the U; are IID(0,62). (i) Let K > 0 be a given number. We wish to estimate B using least-squares...
Question 1(a&b)
Question 3 (a,b,c,d)
QUESTION 1 (15 MARKS) Let X and Y be continuous random variables with joint probability density function 6e.de +3,, х, у z 0 otherwise f(x, y 0 Determine whether or not X and Y are independent. (9 marks) a) b) Find P(x> Y). Show how you get the limits for X and Y (6 marks) QUESTION 3 (19 MARKS) Let f(x, x.) = 2x, , o x, sk: O a) Find k xsl and f(x,...
Question 1 V g(x) O a Let g)-IOdt, where asx Sb. The figure above shows the graph of g on [a.t]. Which of the following could be the graph of f on [a,b]? (B) y (A) Y 0 ct (E) Y D) Question 2 100 12 18 24 Hours The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph shown above. Of the following, which best approximates the total number of...
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