Homework Answers
1. If f(x) is a Density Function, what is the value of k? Skr3, 0<x<1, f(0)...
1. If f(x) is a Density Function, what is the value of k? Skr3, 0<x<1, f(0) 0, elsewhere.
7. For the probability density function f(x) = for 0 <<<2 (a) Find P(x < 1)...
7. For the probability density function f(x) = for 0 <<<2 (a) Find P(x < 1) (b) Find the expected value. (c) Find the variance.
5. Given the probability density f(x)= for -0<x<00, find k. 1+ 2 Jor -
5. Given the probability density f(x)= for -0<x<00, find k. 1+ 2 Jor -
Given f(x) k(1 + x2) 2. -1 < x< 2 a) Find k. b) Calculate F(x)...
Given f(x) k(1 + x2) 2. -1 < x< 2 a) Find k. b) Calculate F(x) and the three quartiles. c) Calculate E(X), Var(X)
2. The beta function B(a, b) is given by B(a,b) = So va-1(1 – v)6–1dv; a...
2. The beta function B(a, b) is given by B(a,b) = So va-1(1 – v)6–1dv; a > 0,6 > 0. The beta distribution has density f(x) = p139–(1 – x)6–1, for 0 < x < 1. If X has the beta distribution, show that E(X) = B(a+1,6)/B(a,b). What is Var(X)?
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find...
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find the marginal probability density function of Y and identify its distribution. (5 marks (b) Find the conditional probability density function of X given Y=y and hence find the mean and variance of X conditional on Y=y. [7 marks] (c) Use iterated expectation to find the expected value of X [5 marks (d) Use E(XY) and var(XY) from (b) above to find the variance of...
A periodic function f(x) with period 21 is defined by: X + -1<x< 0 2 f(x)...
A periodic function f(x) with period 21 is defined by: X + -1<x< 0 2 f(x) = 0<x< 2 Determine the Fourier expansion of the periodic function f(x). X - TT
2. A random variable X has a cdf given by F(x) = 1 . x <...
2. A random variable X has a cdf given by F(x) = 1 . x < 0 0 < x < 1 <3 x > 3 11, (f) What is P(X = 1)? (g) Find E(X), the expectation of X. (h) Find the 75th percentile of the distribution. Namely, find the value of 70.75 SO that P(X < 70.75) = F(710.75) = 0.75. (i) Find the conditional probability P(X > X|X > 3).
(1) Suppose the pdf of a random variable X is 0, otherwise. (a) Find P(2 <...
(1) Suppose the pdf of a random variable X is 0, otherwise. (a) Find P(2 < X < 3). (b) Find P(X < 1). (e) Find t such that P(X <t) = (d) After the value of X has been observed, let y be the integer closest to X. Find the PMF of the random variable y U (2) Suppose for constants n E R and c > 0, we have the function cr" ifa > 1 0, otherwise (a)...
Find the Laplace transform of f(0) = 1, for 0 <t<1 5, for 1<t<2. e-l for...
Find the Laplace transform of f(0) = 1, for 0 <t<1 5, for 1<t<2. e-l for t > 2
1. If f(x) is a Density Function, what is the value of k? Skr3, 0<x<1, f(0) 0, elsewhere.
7. For the probability density function f(x) = for 0 <<<2 (a) Find P(x < 1) (b) Find the expected value. (c) Find the variance.
5. Given the probability density f(x)= for -0<x<00, find k. 1+ 2 Jor -
Given f(x) k(1 + x2) 2. -1 < x< 2 a) Find k. b) Calculate F(x) and the three quartiles. c) Calculate E(X), Var(X)
2. The beta function B(a, b) is given by B(a,b) = So va-1(1 – v)6–1dv; a > 0,6 > 0. The beta distribution has density f(x) = p139–(1 – x)6–1, for 0 < x < 1. If X has the beta distribution, show that E(X) = B(a+1,6)/B(a,b). What is Var(X)?
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find the marginal probability density function of Y and identify its distribution. (5 marks (b) Find the conditional probability density function of X given Y=y and hence find the mean and variance of X conditional on Y=y. [7 marks] (c) Use iterated expectation to find the expected value of X [5 marks (d) Use E(XY) and var(XY) from (b) above to find the variance of...
A periodic function f(x) with period 21 is defined by: X + -1<x< 0 2 f(x) = 0<x< 2 Determine the Fourier expansion of the periodic function f(x). X - TT
2. A random variable X has a cdf given by F(x) = 1 . x < 0 0 < x < 1 <3 x > 3 11, (f) What is P(X = 1)? (g) Find E(X), the expectation of X. (h) Find the 75th percentile of the distribution. Namely, find the value of 70.75 SO that P(X < 70.75) = F(710.75) = 0.75. (i) Find the conditional probability P(X > X|X > 3).
(1) Suppose the pdf of a random variable X is 0, otherwise. (a) Find P(2 < X < 3). (b) Find P(X < 1). (e) Find t such that P(X <t) = (d) After the value of X has been observed, let y be the integer closest to X. Find the PMF of the random variable y U (2) Suppose for constants n E R and c > 0, we have the function cr" ifa > 1 0, otherwise (a)...
Find the Laplace transform of f(0) = 1, for 0 <t<1 5, for 1<t<2. e-l for t > 2
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