Suppose we are trying to model the probability q of winning the lottery using a geometric...
Suppose we are trying to model the probability q of winning the lottery using a geometric distribution. Suppose we have one sample, where a person wins the lottery the second time he plays.
(a) Find a likelihood function L for the parameter q.
(b) Find all critical points of L
(c) Find the maximum likelihood estimator ˆq.
(d) Compute the upper limit ph of the 95% confidence interval for q.
(e) Compute the lower limit pl of the 95% confidence interval for q.
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Suppose we are trying to model the probability q of winning the
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8.40
stion 4 (6 pt) (Ex. 8.40 on page 409 is modified): Suppose that random variable Y is an observation from a normal distribution with unknown mean u and variance l Find and verify a pivotal quantity that you can use to derive confidence limits for the mean u. Find a 95% lower confidence limit for. a. b. 8.40 Suppose that the random variable Yis an observation from a normal distribution with unknown mean μ and variance 1 . Find...
Problem 2. Rice, Problem 7, pg. 314 (Extended)] Suppose that X1,..., Xn iid Geometric(p). a) Find...
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Let X1, ..., X50 denote a random sample of size 50 from the geometric distribution f(x;...
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Question 5-7
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ABayes Is the confidence Region R(a,b) symmetric around the Bayesian estimatorA , as defined earlier? Why or Why not? (...
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Using techniques from Section 8.2, we can find a confidence interval for μd. Consider a random...
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we use a person's dad's height to predict how short or tall the person will be by Suppose building a regression...
we use a person's dad's height to predict how short or tall the person will be by Suppose building a regression model to investigate if a relationship exists between the two variables. Suppose the confidence/prediction interval results are as follows: Predicted/Fitted Values of Height Lower Predicted Bound57.739 Lower Fitted Bound 66.451 Predicted Value Fitted Value 67.532 67.532 Upper Predicted Bound 77.326 Upper Fitted Bound 68.613 SE (Fitted Value) SE (Predicted Value) 0.5432 4.9212 Unusualness (Leverage) 0.0123 Percent Coverage 95 Corresponding...
Suppose observations X1, X2,.. are recorded. We assume these to be conditionally independent and exponen- tially...
Suppose observations X1, X2,.. are recorded. We assume these to be conditionally independent and exponen- tially distributed given a parameter θ: Xi ~' Exponential(θ), for all i 1, . . . , n. The exponential distribution is controlled by one rate parameter θ > 0, and its density is for r ER+ 1. Plot the graph of p(x:0) for θ 1 in the interval x E [0,4] 2. What is the visual representation of the likelihood of individual data points?...
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R CODE PROGRAM 1. Suppose we want to simulate an experiment that can take outcomes 1; : : : ; n with probabilies p1; : : : ; pn. To be specic, suppose the R-vector p=c(.1,.2,.3,.35, .02, .03) gives the desired probabilities. Write R code that produces a number from 1 to 6 with the given probabilities, without using if statements. I recommend using the R command cumsum to do this, though there many possible approaches. 2. Suppose we are...
8.40
stion 4 (6 pt) (Ex. 8.40 on page 409 is modified): Suppose that random variable Y is an observation from a normal distribution with unknown mean u and variance l Find and verify a pivotal quantity that you can use to derive confidence limits for the mean u. Find a 95% lower confidence limit for. a. b. 8.40 Suppose that the random variable Yis an observation from a normal distribution with unknown mean μ and variance 1 . Find...
Problem 2. Rice, Problem 7, pg. 314 (Extended)] Suppose that X1,..., Xn iid Geometric(p). a) Find the method of moments estimator for p. (b) Find the maximum likelihood estimator for p. (c) Find the asymptotic variance of the MLE (d) Suppose that p has a uniform prior distribution on the interval [0, 1]. What is the posterior distribution of p? For part (e), assume that we obtained a random sample of size 4 with L^^^xi-.4 (e) What is the posterior...
Question 5-7
2 Gamma waiting times (frequentist) Suppose we model a sample of times between arrivals of the 1 train of the New York City subway at the 116th Street station, y1, . . . , Yn, as IID random variables Y1, ... , Yn sampled from a Gam(v, 1) distribution, for some unknown v and X. 1. What is the joint log-likelihood, In fy ...Y|0,1(91, ... , Yn | v, 4)? [5 mark(s)] 2. For a fixed value of...
ABayes Is the confidence Region R(a,b) symmetric around the Bayesian estimatorA , as defined earlier? Why or Why not? (a)Yes, because by construction, confidence intervals and hence confidence regions are symmetric around any consistent of the parameter (b)Yes, because our posterior distribution is symmetric and we chose a and b such that (-o0, a) and (b,oo) have an equal 5% Probability. (c)No, because our posterior distribution is not symmetric (it is either skewed to the left or skewed to the...
we use a person's dad's height to predict how short or tall the person will be by Suppose building a regression model to investigate if a relationship exists between the two variables. Suppose the confidence/prediction interval results are as follows: Predicted/Fitted Values of Height Lower Predicted Bound57.739 Lower Fitted Bound 66.451 Predicted Value Fitted Value 67.532 67.532 Upper Predicted Bound 77.326 Upper Fitted Bound 68.613 SE (Fitted Value) SE (Predicted Value) 0.5432 4.9212 Unusualness (Leverage) 0.0123 Percent Coverage 95 Corresponding...
Suppose observations X1, X2,.. are recorded. We assume these to be conditionally independent and exponen- tially distributed given a parameter θ: Xi ~' Exponential(θ), for all i 1, . . . , n. The exponential distribution is controlled by one rate parameter θ > 0, and its density is for r ER+ 1. Plot the graph of p(x:0) for θ 1 in the interval x E [0,4] 2. What is the visual representation of the likelihood of individual data points?...
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