Homework Answers
OR = P(disease|Exposed)/ (1-P(disease|Exposed)) /
(P(disease|Unexposed)/ (1-P(disease|Unexposed)) )
= (a/(a+c))/(c/(a+c)) / (b/(b+d))/(d/(b+d))
=(a/c)/(b/d)
= ad /(b*c)
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Suppose we have a sample of observations for the pair of random variable (X, Y) in the following ...
Let the random variable Y represent hourly wages and the random variable X represent education. Suppose we have the...
Let the random variable Y represent hourly wages and the random variable X represent education. Suppose we have the following regression equation in mind to estimate the return to education: (a) Can we say that this regression would capture the causal effect of education on wages? Support your answer with reasoning. (b) Using the sample equivalent of the two equations E(u)-0 and E(uX)-0 derive the regression estimators for A, and β1-Write down each mathe tnatical step, what would be the...
3. Suppose we have a random variable X with mean a new random variable Y as...
3. Suppose we have a random variable X with mean a new random variable Y as = 7 and variance a4. We define Y 3 5X Find the standard deviation of Y
2. Given the following model: Y, = B. +X;B, + Mi a. Suppose we estimate the...
2. Given the following model: Y, = B. +X;B, + Mi a. Suppose we estimate the model ignoring the constant term. Show that the resulting estimator (call it ß, ) is biased. b. Derive the variance
4. Let X be a random variable with pdf f(x). Suppose that the mean of X...
4. Let X be a random variable with pdf f(x). Suppose that the mean of X is 2 and the variance of X is 5. It is easy to show that the pdf of Y = 0X is fo(y) = f(1/0) (You do not have to show this, but it's good practice.) Suppose the popula- tion has the distribution of foly) with 8 unknown. We take a random sample {Y}}=1 and compute the sample mean Y. (a) What is a...
Let X be a random variable with cdf FX (x:0), expected value EIX-μ and variance VlX- σ2. Let X1,X...
Let X be a random variable with cdf FX (x:0), expected value EIX-μ and variance VlX- σ2. Let X1,X2, , Xn be an id sample drawn according to FX(x,8) where Fx (x,8) =万 for all x E (0,0). Let max(X1, X2, , X.) be an estimator of θ, suggested from pure common sense. Remember that if Y = max(X1, X2, , Xn). Then it can be shown that the cdf Fy () of Y is given by Fr(u) (Fx()" where...
49. Suppose that N e Po(A) independent observations of a random variable, X, with mean 0...
49. Suppose that N e Po(A) independent observations of a random variable, X, with mean 0 and variance 1, are is independent of X1, X2, ... . Show that performed. Moreover, assume that N X1X2 XN VN d N(0, 1) as
49. Suppose that N e Po(A) independent observations of a random variable, X, with mean 0 and variance 1, are is independent of X1, X2, ... . Show that performed. Moreover, assume that N X1X2 XN VN d N(0,...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
1. Consider a variable y = θ+e where θ is an unknown parameter and e is a random variable with me...
1. Consider a variable y = θ+e where θ is an unknown parameter and e is a random variable with mean zero (a) What is the expected value of y (b) Suppose you draw a sample of in y-Derive the least squares estimator for θ. For full credit you must check the 2nd order condition. (c) Can this estimator () be described as a method of moments estimator? (d) Now suppose e is independent normally distributed with mean 0 and...
I. Consider a variable y = θ + where θ is an unknown parameter and e is a random variable with me...
I. Consider a variable y = θ + where θ is an unknown parameter and e is a random variable with mean zero. (a) What is the expected value of y? (b) Suppose you draw a sample of yi yn. Derive the least squares estimator for θ. For full credit you must check the 2nd order condition c) Can this estimator (0) be described as a method of moments estimator? (d) Now suppose є is independent normally distributed with mean...
2. Suppose XX2,X is a random sample from an exponential distribution with . Let X(1) minX1,X2,...
2. Suppose XX2,X is a random sample from an exponential distribution with . Let X(1) minX1,X2, Xn), the minimum of the sample mean (a) Show that the estimator 6nx is an unbiased estimator of 8. (hint: you were asked to derive the distribution of X for a random sample from an exponential distribution on assignment 2 -you may use the result) (b) X, the sample mean, is also an unbiased estimator of . Which of the unbiased estimators, or X,...
Let the random variable Y represent hourly wages and the random variable X represent education. Suppose we have the following regression equation in mind to estimate the return to education: (a) Can we say that this regression would capture the causal effect of education on wages? Support your answer with reasoning. (b) Using the sample equivalent of the two equations E(u)-0 and E(uX)-0 derive the regression estimators for A, and β1-Write down each mathe tnatical step, what would be the...
3. Suppose we have a random variable X with mean a new random variable Y as = 7 and variance a4. We define Y 3 5X Find the standard deviation of Y
2. Given the following model: Y, = B. +X;B, + Mi a. Suppose we estimate the model ignoring the constant term. Show that the resulting estimator (call it ß, ) is biased. b. Derive the variance
4. Let X be a random variable with pdf f(x). Suppose that the mean of X is 2 and the variance of X is 5. It is easy to show that the pdf of Y = 0X is fo(y) = f(1/0) (You do not have to show this, but it's good practice.) Suppose the popula- tion has the distribution of foly) with 8 unknown. We take a random sample {Y}}=1 and compute the sample mean Y. (a) What is a...
Let X be a random variable with cdf FX (x:0), expected value EIX-μ and variance VlX- σ2. Let X1,X2, , Xn be an id sample drawn according to FX(x,8) where Fx (x,8) =万 for all x E (0,0). Let max(X1, X2, , X.) be an estimator of θ, suggested from pure common sense. Remember that if Y = max(X1, X2, , Xn). Then it can be shown that the cdf Fy () of Y is given by Fr(u) (Fx()" where...
49. Suppose that N e Po(A) independent observations of a random variable, X, with mean 0 and variance 1, are is independent of X1, X2, ... . Show that performed. Moreover, assume that N X1X2 XN VN d N(0, 1) as
49. Suppose that N e Po(A) independent observations of a random variable, X, with mean 0 and variance 1, are is independent of X1, X2, ... . Show that performed. Moreover, assume that N X1X2 XN VN d N(0,...
For observations {Y, X;}=1, recall that for the model Y = 0 + Box: +e the OLS estimator for {00, Bo}, the minimizer of E. (Y: - a - 3x), is . (X.-X) (Y-Y) and a-Y-3X. - (Xi - x) When the equation (1) is the true data generating process, {X}- are non-stochastic, and {e} are random variables with B (ei) = 0, B(?) = 0, and Ele;e;) = 0 for any i, j = 1,2,...,n and i j, we...
1. Consider a variable y = θ+e where θ is an unknown parameter and e is a random variable with mean zero (a) What is the expected value of y (b) Suppose you draw a sample of in y-Derive the least squares estimator for θ. For full credit you must check the 2nd order condition. (c) Can this estimator () be described as a method of moments estimator? (d) Now suppose e is independent normally distributed with mean 0 and...
I. Consider a variable y = θ + where θ is an unknown parameter and e is a random variable with mean zero. (a) What is the expected value of y? (b) Suppose you draw a sample of yi yn. Derive the least squares estimator for θ. For full credit you must check the 2nd order condition c) Can this estimator (0) be described as a method of moments estimator? (d) Now suppose є is independent normally distributed with mean...
2. Suppose XX2,X is a random sample from an exponential distribution with . Let X(1) minX1,X2, Xn), the minimum of the sample mean (a) Show that the estimator 6nx is an unbiased estimator of 8. (hint: you were asked to derive the distribution of X for a random sample from an exponential distribution on assignment 2 -you may use the result) (b) X, the sample mean, is also an unbiased estimator of . Which of the unbiased estimators, or X,...
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