Suppose we wish to predict the value of a random variable X by using one of...
Suppose we wish to predict the value of a random variable X by using one of the predictors Y1, Y2 , …, Yn, each of which satisfies E[Yi | X] = X. Show that the predictor Yi that minimizes E[(Yi − X) 2 ] is the one whose variance is smallest
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Suppose we wish to predict the value of a random variable X by
using one of...
2. [x] Suppose that Y1, Y2, Y3 denote a random sample from an exponential distribution whose...
2. [x] Suppose that Y1, Y2, Y3 denote a random sample from an exponential distribution whose pdf and cdf are given by f(y) = (1/0)e¬y/® and F(y) =1 – e-y/0, 0 > 0. It is also known that E[Y;] = 0. ', y > 0, respectively, with some unknown (a) Let X = min{Y1,Y2, Y3}. Show that X has pdf given by f(æ) = (3/0)e-3y/º. Start by thinking about 1- F(x) = Pr(min{Y1,Y2, Y3} > x) = Pr(Y1 > x,...
PART B: Application 5. Suppose that you observe a random variable X. and then, on the...
PART B: Application 5. Suppose that you observe a random variable X. and then, on the basis of the observed value. you attempt to predict the value of a second random variable Y. Let Y denote the predictor or an estimator of Y ; that is, if X is observed to equal , then Y is your prediction for the value of Y, and your goal is to choose Y so that it tends to be close to Y First,...
Question 3: A random variable X has a Bernoulli distribution with parameter θ є (0,1) if...
Question 3: A random variable X has a Bernoulli distribution with parameter θ є (0,1) if X {0,1} and P(X-1)-θ. Suppose that we have nd random variables y, x, following a Bernoulli(0) distribution and observed values y1,... . Jn a) Show that EIX) θ and Var[X] θ(1-0). b) Let θ = ỹ = (yit . .-+ yn)/n. Show that θ is unbiased for θ and compute its variance. c) Let θ-(yit . . . +yn + 1)/(n + 2) (this...
1. Suppose X1, ..., Xn be a random sample from Exp(1) and Y1 < ... <...
1. Suppose X1, ..., Xn be a random sample from Exp(1) and Y1 < ... < Yn be the order statistics from this sample. a) Find the joint pdf of (Y1, .. , Yn). b) Find the joint pdf of (W1, .. , Wn) where W1 = nY1, W2 = (n-1)(Y2 -Y1), W3 = (n - 2)(Y3 - Y2),..., Wn-1 = 2(Yn-1 - Yn-2), Wn = Yn - Yn-1. (c) Show that Wi's are independent and its distribution is identically...
A. We wish to etima the anea a R of a rectangular po of lan, with length μ1 and width Ha. We meas...
A. We wish to etima the anea a R of a rectangular po of lan, with length μ1 and width Ha. We measure the length and the width twice. There are two natural ways to estimate the unknown constant a. We can either multiply the average width and length, or we can take the average of the two estimated areas. Suppose the measurements are outcomes of independent random variables Xi, X2 ~ Nga, σ*) and Y1-Y2 ~ N412, σ2 )....
3. Use Matlab to check that if X is a Beta-(a,b) random variable with a =...
3. Use Matlab to check that if X is a Beta-(a,b) random variable with a = b = į and if, conditional on this Y1, Y2 are independent Bernoulli-p random variables for p = X, and we find Yi + Y2 = k = 1 (so that 2 - Y1 - Y2 is also 1), then the conditional distributrion of X is Beta-(a', b') for a' = b' = 3. My plot (for the number of trials equal to 100,000)...
Use this result without proof: if X and Y are two normal random variables with means...
Use this result without proof: if X and Y are two normal random variables with means ux and My respectively, and variances oź and oſ respectively, and Z = X+Y, Z is also a normal random variable with mean (ux + Hy) and variance (ox +og). a) Suppose Yı, Y2, Yz, Y4 and Y5 are all independent normal random variables, each with a mean of 1 and a variance of 5. What is the probability that (Y1 + 2Y2 +...
Suppose we have a sample of observations for the pair of random variable (X, Y) in the following ...
Suppose we have a sample of observations for the pair of random variable (X, Y) in the following 2 x2 Show that the odds ratio can be estimated by ad/bc and derive an estimate of the variance of this estimator
Suppose we have a sample of observations for the pair of random variable (X, Y) in the following 2 x2 Show that the odds ratio can be estimated by ad/bc and derive an estimate of the variance of this estimator
Suppose that we wish to predict whether a given stock will issue a dividend this year...
Suppose that we wish to predict whether a given stock
will issue a dividend this year (`Yes'or `No') based on X, last
year's percent profit.
Suppose that we wish to predict whether a given stock will issue a dividend this year ('Yes' or 'No') based on X, last year's percent profit. We examine a large number of companies and discover that X- 12 for dividend providing companies, while the mean for those that didn't was X 1.5. In addition, the...
Question 3 [25] , Yn denote a random sample of size n from a Let Y,...
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...
2. [x] Suppose that Y1, Y2, Y3 denote a random sample from an exponential distribution whose pdf and cdf are given by f(y) = (1/0)e¬y/® and F(y) =1 – e-y/0, 0 > 0. It is also known that E[Y;] = 0. ', y > 0, respectively, with some unknown (a) Let X = min{Y1,Y2, Y3}. Show that X has pdf given by f(æ) = (3/0)e-3y/º. Start by thinking about 1- F(x) = Pr(min{Y1,Y2, Y3} > x) = Pr(Y1 > x,...
PART B: Application 5. Suppose that you observe a random variable X. and then, on the basis of the observed value. you attempt to predict the value of a second random variable Y. Let Y denote the predictor or an estimator of Y ; that is, if X is observed to equal , then Y is your prediction for the value of Y, and your goal is to choose Y so that it tends to be close to Y First,...
Question 3: A random variable X has a Bernoulli distribution with parameter θ є (0,1) if X {0,1} and P(X-1)-θ. Suppose that we have nd random variables y, x, following a Bernoulli(0) distribution and observed values y1,... . Jn a) Show that EIX) θ and Var[X] θ(1-0). b) Let θ = ỹ = (yit . .-+ yn)/n. Show that θ is unbiased for θ and compute its variance. c) Let θ-(yit . . . +yn + 1)/(n + 2) (this...
A. We wish to etima the anea a R of a rectangular po of lan, with length μ1 and width Ha. We measure the length and the width twice. There are two natural ways to estimate the unknown constant a. We can either multiply the average width and length, or we can take the average of the two estimated areas. Suppose the measurements are outcomes of independent random variables Xi, X2 ~ Nga, σ*) and Y1-Y2 ~ N412, σ2 )....
3. Use Matlab to check that if X is a Beta-(a,b) random variable with a = b = į and if, conditional on this Y1, Y2 are independent Bernoulli-p random variables for p = X, and we find Yi + Y2 = k = 1 (so that 2 - Y1 - Y2 is also 1), then the conditional distributrion of X is Beta-(a', b') for a' = b' = 3. My plot (for the number of trials equal to 100,000)...
Use this result without proof: if X and Y are two normal random variables with means ux and My respectively, and variances oź and oſ respectively, and Z = X+Y, Z is also a normal random variable with mean (ux + Hy) and variance (ox +og). a) Suppose Yı, Y2, Yz, Y4 and Y5 are all independent normal random variables, each with a mean of 1 and a variance of 5. What is the probability that (Y1 + 2Y2 +...
Suppose we have a sample of observations for the pair of random variable (X, Y) in the following 2 x2 Show that the odds ratio can be estimated by ad/bc and derive an estimate of the variance of this estimator
Suppose we have a sample of observations for the pair of random variable (X, Y) in the following 2 x2 Show that the odds ratio can be estimated by ad/bc and derive an estimate of the variance of this estimator
Suppose that we wish to predict whether a given stock
will issue a dividend this year (`Yes'or `No') based on X, last
year's percent profit.
Suppose that we wish to predict whether a given stock will issue a dividend this year ('Yes' or 'No') based on X, last year's percent profit. We examine a large number of companies and discover that X- 12 for dividend providing companies, while the mean for those that didn't was X 1.5. In addition, the...
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...
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