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1. (Distributions with Random Parameters) Suppose that the density X of red blood corpuscles in h...
A random varible X taking values from [0,1] has Beta distribution of parameters a and B,...
A random varible X taking values from [0,1] has Beta distribution of parameters a and B, which we denote by Beta(a,b), if it has PDF _f(a+B) fa-1(1 – X)B-1, fx(x) = T(a)l(B) where I(z) is the Euler Gamma function defined by I(z) = Sx2-1e-*dx. Bob has a coin with unknown probability of heads. Alice has the following Beta prior: A = Beta(2,3). Suppose that Bob gives Alice the data on = {x1,...,xn), which is the outcome of n indepen- dent...
Problem 6. [Poisson is Pronounced 'Pwah-ssohn] (a) Suppose that X is a random variable following the...
Problem 6. [Poisson is Pronounced 'Pwah-ssohn] (a) Suppose that X is a random variable following the Poisson distribution with rate parameter A. Show that E[x]-A Hint: You may find the following fact useful: at k! (b) Suppose that we obtained the following count data: Count Frequency 24 30 17 19 Fit a Poisson distribution to the data using the Method of Moments (c) Suppose that X is a random variable that follows the Poisson distribution that you fit in part...
Problem 1: 10 points Assume that a random variable X follows the Poisson distribution with intensity-A,...
Problem 1: 10 points Assume that a random variable X follows the Poisson distribution with intensity-A, that is k! for k 0,1,2, . Using the identity (valid for all real t) exp(t) = Σ冠. k! k=0 derive the probability that X takes an even value, that is PIX is even
P7 continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The...
P7
continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
Question 3 Suppose that the random variable X has the Poisson distribution, with P (X0) 0.4....
Question 3 Suppose that the random variable X has the Poisson distribution, with P (X0) 0.4. (a) Calculate the probability P (X <3) (b) Calculate the probability P (X-0| X <3) (c) Prove that Y X+1 does not have the Polsson distribution, by calculating P (Y0) Question 4 The random variable X is uniformly distributed on the interval (0, 2) and Y is exponentially distrib- uted with parameter λ (expected value 1 /2). Find the value of λ such that...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable X follows a Poisson distribution with parameter λ if Pr(X = k) = Ake-A ke(0, 1,2, ) k! You are a warrior in Peter Jackson's The Hobbit: Battle of the Five Armies. Because Peter decided to make his battle scenes as legendary as possible, he's decided that the number of orcs that will die with one swing of your sword is Poisson distributed (lid)...
Problem 1 (16 points). Suppose that X and Y are independent random variables and that Y...
Problem 1 (16 points). Suppose that X and Y are independent random variables and that Y follows a geometric distribution with parameter p. Assume that X takes only nonnegative integer values, and let Gx(z) be the probability generating function of X. (We make no additional assumptions about the distribution of X.) Show that P(X<Y) = Gx(1- p). Clearly indicate the step(s) in your argument that use the assumption that X and Y are independent.
Question 1: 1a) Let the random variable X have a geometric distribution with parameter p ,...
Question 1: 1a) Let the random variable X have a geometric distribution with parameter p , i.e., P(X = x) = pq??, x=1,2,... i) Show that P(X > m)=q" , where m is a positive integer. (5 points) ii) Show that P(X > m+n X > m) = P(X>n), where m and n are positive integers. (5 points) 1b) Suppose the random variable X takes non-negative integer values, i.e., X is a count random variable. Prove that (6 points) E(X)=...
This is for an Information Theory class. H(X) is entropy rate. Problem 8: Suppose that X is a random variable with a pr...
This is for an Information Theory class. H(X) is entropy
rate.
Problem 8: Suppose that X is a random variable with a probability that X = k) given by: probability distribution (i.e., Px (k) = Prob(X = k) = (1 - ) )X* for k0 where 0 < 1 and k is a non-negative integer (and hence X can take any negative integer value). To answer this question, note that the AEP theorem we proved for a finite-alphabet random variable...
1) Suppose that three random variables, X, Y, and Z have a continuous joint probability density...
1) Suppose that three random variables, X, Y, and Z have a continuous joint probability density function f(x, y. z) elsewhere a) Determine the value of the constant b) Find the marginal joint p. d. fof X and Y, namely f(x, y) (3 Points) c) Using part b), compute the conditional probability of Z given X and Y. That is, find f (Z I x y) d) Using the result from part c), compute P(Z<0.5 x - 3 Points) 2...
A random varible X taking values from [0,1] has Beta distribution of parameters a and B, which we denote by Beta(a,b), if it has PDF _f(a+B) fa-1(1 – X)B-1, fx(x) = T(a)l(B) where I(z) is the Euler Gamma function defined by I(z) = Sx2-1e-*dx. Bob has a coin with unknown probability of heads. Alice has the following Beta prior: A = Beta(2,3). Suppose that Bob gives Alice the data on = {x1,...,xn), which is the outcome of n indepen- dent...
Problem 6. [Poisson is Pronounced 'Pwah-ssohn] (a) Suppose that X is a random variable following the Poisson distribution with rate parameter A. Show that E[x]-A Hint: You may find the following fact useful: at k! (b) Suppose that we obtained the following count data: Count Frequency 24 30 17 19 Fit a Poisson distribution to the data using the Method of Moments (c) Suppose that X is a random variable that follows the Poisson distribution that you fit in part...
Problem 1: 10 points Assume that a random variable X follows the Poisson distribution with intensity-A, that is k! for k 0,1,2, . Using the identity (valid for all real t) exp(t) = Σ冠. k! k=0 derive the probability that X takes an even value, that is PIX is even
P7
continuous random variable X has the probability density function fx(x) = 2/9 if P.5 The absolutely continuous random 0<r<3 and 0 elsewhere). Let (1 - if 0<x< 1, g(x) = (- 1)3 if 1<x<3, elsewhere. Calculate the pdf of Y = 9(X). P. 6 The absolutely continuous random variables X and Y have the joint probability density function fx.ya, y) = 1/(x?y?) if x > 1,y > 1 (and 0 elsewhere). Calculate the joint pdf of U = XY...
Question 3 Suppose that the random variable X has the Poisson distribution, with P (X0) 0.4. (a) Calculate the probability P (X <3) (b) Calculate the probability P (X-0| X <3) (c) Prove that Y X+1 does not have the Polsson distribution, by calculating P (Y0) Question 4 The random variable X is uniformly distributed on the interval (0, 2) and Y is exponentially distrib- uted with parameter λ (expected value 1 /2). Find the value of λ such that...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable X follows a Poisson distribution with parameter λ if Pr(X = k) = Ake-A ke(0, 1,2, ) k! You are a warrior in Peter Jackson's The Hobbit: Battle of the Five Armies. Because Peter decided to make his battle scenes as legendary as possible, he's decided that the number of orcs that will die with one swing of your sword is Poisson distributed (lid)...
Problem 1 (16 points). Suppose that X and Y are independent random variables and that Y follows a geometric distribution with parameter p. Assume that X takes only nonnegative integer values, and let Gx(z) be the probability generating function of X. (We make no additional assumptions about the distribution of X.) Show that P(X<Y) = Gx(1- p). Clearly indicate the step(s) in your argument that use the assumption that X and Y are independent.
Question 1: 1a) Let the random variable X have a geometric distribution with parameter p , i.e., P(X = x) = pq??, x=1,2,... i) Show that P(X > m)=q" , where m is a positive integer. (5 points) ii) Show that P(X > m+n X > m) = P(X>n), where m and n are positive integers. (5 points) 1b) Suppose the random variable X takes non-negative integer values, i.e., X is a count random variable. Prove that (6 points) E(X)=...
This is for an Information Theory class. H(X) is entropy
rate.
Problem 8: Suppose that X is a random variable with a probability that X = k) given by: probability distribution (i.e., Px (k) = Prob(X = k) = (1 - ) )X* for k0 where 0 < 1 and k is a non-negative integer (and hence X can take any negative integer value). To answer this question, note that the AEP theorem we proved for a finite-alphabet random variable...
1) Suppose that three random variables, X, Y, and Z have a continuous joint probability density function f(x, y. z) elsewhere a) Determine the value of the constant b) Find the marginal joint p. d. fof X and Y, namely f(x, y) (3 Points) c) Using part b), compute the conditional probability of Z given X and Y. That is, find f (Z I x y) d) Using the result from part c), compute P(Z<0.5 x - 3 Points) 2...
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