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Problem 1. (5 marks; 2, 3) Assume that UUniform(0,1 Answer each of the following questions. You...
Question 3 [17 marks] The random variable X is distributed exponentially with parameter A i.e. X~...
Question 3 [17 marks] The random variable X is distributed exponentially with parameter A i.e. X~ Exp(A), so that its probability density function (pdf) of X is SO e /A fx(x) | 0, (2) (a) Let Y log(X. When A = 1, (i) Show that the pdf of Y is fr(y) = e (u+e-") (ii) Derive the moment generating function of Y, My(t), and give the values of t such that My(t) is well defined. (b) Suppose that Xi, i...
Problem 1. (5 marks. 3. 2) Assume X ~ Gamma(01, β) and Y ~ Gamma(O2, β)...
Problem 1. (5 marks. 3. 2) Assume X ~ Gamma(01, β) and Y ~ Gamma(O2, β) are independent random variables. a) Compute the Joint density of U = X + Y and V X X + Y , be sure to include the support/domain. b) Based on the joint density derived in part (a) find the marginal densities of U and V, be sure to include the support (s)/domain(s). Explicitly state the name of the distributions of U and V...
In question 5, f(x) = λ*exp(-λx), for x greater or equal to 0, and zero otherwise....
In question 5, f(x) = λ*exp(-λx), for x greater or
equal to 0, and zero otherwise.
9. Let X have an exponential distribution with λ = 1 (see Question 5), and let Y = log(X). Find the probability density function of Y. Where is the density non-zero? Note that in this course, log refers to the log base e, or natural log, often symbolized In. The distribution of Y is called the (standard) Gumbel, or extreme value distribution. 2
12.5A e 2 Suppose that A has a Gamma distribution: fA(A) 「3.5)23.5 (a) Suppose that the...
12.5A e 2 Suppose that A has a Gamma distribution: fA(A) 「3.5)23.5 (a) Suppose that the conditional distribution of X given Λ = λ is fxA(TA z ) e- for x > 0. i. Find Ex ii. Find Var( (b) Suppose that the conditional distribution of X given A = λ is frA (zA)-Xe-k for x > 0. Find the unconditional probability density function fx(x) of *
1. The size of claims made on an insurance policy are modelled through the following distribu- ti...
1. The size of claims made on an insurance policy are modelled through the following distribu- tion: You are interested in estimating the parameter λ > 0, using the following observations: 120, 20, 60, 70, 110, 150, 220, 160, 100, 100 (a) Verify that f is a density (b) Find the expectation of the generic random variable X, as a function of \ when A 1 (c) Prove that the method of moments estimator of λ is λι =斉. Calculate...
Section B (50 marks) Answer any two questions from this section. All questions carry equal marks....
Section B (50 marks) Answer any two questions from this section. All questions carry equal marks. 1. (a) The density function of X is given by $(x) = { a + bx?, OSX31 0, otherwise If E[X] = , find i. the cumulative density function, F(X). [5 mark ii. P(x < [3 mark (b) Given the observations Ii, i = 1, 2 and their corresponding weight, W;, i = 1, show that the weighted average Xw = 221 + (1...
3. Suppose that the 5-year survival probability, X, for women with breast cancer who live in...
3. Suppose that the 5-year survival probability, X, for women with breast cancer who live in a rural county follows Beta distribution with probability density function (pdf) fx (20) = 0.00-1 where 0 < x < 1 and parameter 6 > 0. Let X1, ..., X, be a random sample of size n from a population of rural counties. Researchers intend to make statistical inference on the parameter 6 using collected data X1, ..., (a) Let Y; = – log(Xi)...
(20pts) 1. For this problem, answer the following questions. (a) State the domain and range of...
(20pts) 1. For this problem, answer the following questions. (a) State the domain and range of the following function f(x) = -55:<2 1, 2sr35 (i) Domain (ii) Range = (b) Evaluate (1)f(-3) = (ii) f(2)= (iii) (4) = (c) Given the followig augumented matrix write a system of equations it represents. Assume the variables are r. v. Do not solve. (d) What is the solution for the system of equations equivalent to the following reduces augmented matrix? Assume the variables...
(1 point) If the joint density function of X and Y is f(x, y) = c(22...
(1 point) If the joint density function of X and Y is f(x, y) = c(22 - y2)e- with OS: < oo and I y I, find each of the following. (a) The conditional probability density of X given Y = y >0. Conditional density fxy(:, y) = (Enter your answer as a function of I, with y as a parameter.) (b) The conditional probability distribution of Y given X = 2. Conditional distribution Fyx (2) = (Enter your answer...
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...
Question 3 [17 marks] The random variable X is distributed exponentially with parameter A i.e. X~ Exp(A), so that its probability density function (pdf) of X is SO e /A fx(x) | 0, (2) (a) Let Y log(X. When A = 1, (i) Show that the pdf of Y is fr(y) = e (u+e-") (ii) Derive the moment generating function of Y, My(t), and give the values of t such that My(t) is well defined. (b) Suppose that Xi, i...
Problem 1. (5 marks. 3. 2) Assume X ~ Gamma(01, β) and Y ~ Gamma(O2, β) are independent random variables. a) Compute the Joint density of U = X + Y and V X X + Y , be sure to include the support/domain. b) Based on the joint density derived in part (a) find the marginal densities of U and V, be sure to include the support (s)/domain(s). Explicitly state the name of the distributions of U and V...
In question 5, f(x) = λ*exp(-λx), for x greater or
equal to 0, and zero otherwise.
9. Let X have an exponential distribution with λ = 1 (see Question 5), and let Y = log(X). Find the probability density function of Y. Where is the density non-zero? Note that in this course, log refers to the log base e, or natural log, often symbolized In. The distribution of Y is called the (standard) Gumbel, or extreme value distribution. 2
12.5A e 2 Suppose that A has a Gamma distribution: fA(A) 「3.5)23.5 (a) Suppose that the conditional distribution of X given Λ = λ is fxA(TA z ) e- for x > 0. i. Find Ex ii. Find Var( (b) Suppose that the conditional distribution of X given A = λ is frA (zA)-Xe-k for x > 0. Find the unconditional probability density function fx(x) of *
1. The size of claims made on an insurance policy are modelled through the following distribu- tion: You are interested in estimating the parameter λ > 0, using the following observations: 120, 20, 60, 70, 110, 150, 220, 160, 100, 100 (a) Verify that f is a density (b) Find the expectation of the generic random variable X, as a function of \ when A 1 (c) Prove that the method of moments estimator of λ is λι =斉. Calculate...
Section B (50 marks) Answer any two questions from this section. All questions carry equal marks. 1. (a) The density function of X is given by $(x) = { a + bx?, OSX31 0, otherwise If E[X] = , find i. the cumulative density function, F(X). [5 mark ii. P(x < [3 mark (b) Given the observations Ii, i = 1, 2 and their corresponding weight, W;, i = 1, show that the weighted average Xw = 221 + (1...
3. Suppose that the 5-year survival probability, X, for women with breast cancer who live in a rural county follows Beta distribution with probability density function (pdf) fx (20) = 0.00-1 where 0 < x < 1 and parameter 6 > 0. Let X1, ..., X, be a random sample of size n from a population of rural counties. Researchers intend to make statistical inference on the parameter 6 using collected data X1, ..., (a) Let Y; = – log(Xi)...
(20pts) 1. For this problem, answer the following questions. (a) State the domain and range of the following function f(x) = -55:<2 1, 2sr35 (i) Domain (ii) Range = (b) Evaluate (1)f(-3) = (ii) f(2)= (iii) (4) = (c) Given the followig augumented matrix write a system of equations it represents. Assume the variables are r. v. Do not solve. (d) What is the solution for the system of equations equivalent to the following reduces augmented matrix? Assume the variables...
(1 point) If the joint density function of X and Y is f(x, y) = c(22 - y2)e- with OS: < oo and I y I, find each of the following. (a) The conditional probability density of X given Y = y >0. Conditional density fxy(:, y) = (Enter your answer as a function of I, with y as a parameter.) (b) The conditional probability distribution of Y given X = 2. Conditional distribution Fyx (2) = (Enter your answer...
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...
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