![Suppose two treatments will be given to n patients, randomly sampled from a population. Let uet 1-{ 2 otheatme t one-topat enti, 1 if treatment one is given to patient i 2 otherwise. Let the response be r-{ 1 if treatment t cure patient i, i0 otherwise Here thechance of patient i being given Ti = 1 is 0.5. We want to estimate ㎡ = P07 = 1) for t = 1,2. Assume that (Yin, Ti); i-1 ,n, are independent and identically distributed, and the prior distribution on (π1, π2) is uniform on [0, 1] × [0, 1]. Calculate the posterior density of P((π, π*)!(Y1Ti , . . . , YT, Tİ , , , , , TOD](http://img.homeworklib.com/questions/975a18b0-74bb-11ea-85e9-1d4898b36ff1.png?x-oss-process=image/resize,w_560)
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Suppose two treatments will be given to n patients, randomly sampled from a population. Let uet...
Suppose two treatments will be given to n patients, randomly sampled from a population. Let uet...
Suppose two treatments will be given to n patients, randomly sampled from a population. Let uet 1-{ 2 otheatme t one-topat enti, 1 if treatment one is given to patient i 2 otherwise. Let the response be r-{ 1 if treatment t cure patient i, i0 otherwise Here thechance of patient i being given Ti = 1 is 0.5. We want to estimate ㎡ = P07 = 1) for t = 1,2. Assume that (Yin, Ti); i-1 ,n, are independent...
Please help! (a) Let us consider a full model of a balanced (all t treatments have...
Please help!
(a) Let us consider a full model of a balanced (all t treatments have equal number of observations r) CRD design with t treatments and r replications of each treatment, hence having n-rt observations. 2. i. Minimizing sum of square error Δfull (μ'Ti) -Σι-1 Σ-1 (Vi,- μ-Ti)2 with respect to μ and Ti find the least square estimators of μ and Ti as μ and Ti. Hint: Take derivative of the objective function with respect to μ and...
2. (a) Let us consider a full model of a balanced (all t treatments have equal...
2. (a) Let us consider a full model of a balanced (all t treatments have equal number of observations r) CRD design with t treatments and r replications of each treatment, hence having n-rt observations i. Minimizing sum of square error Δfull(μ, Tỉ)-Σι-12jai (Vij-l-ri)2 with respect to μ and Ti find the least square estimators of μ and Te as μ and Ti Hint: Take derivative of the objective function with respect to u and Ti and equate then to...
1. Suppose we are going to sample n individuals and ask each sampled person whether they...
1. Suppose we are going to sample n individuals and ask each sampled person whether they support policy A or not. Let Yi Y0 otherwise 1 if person i in the sample supports the policy, and (a) Assume Y1, , Yn are, conditional on θ. 1.1.d. binary random variables with expec- tation θ. Write down the joint distribution Pr(Yi-yi, . . . ,Ý,-yn(9) in a compact form. Also write down the form of Pr(> Ý,-y|0) (b) For the moment, suppose...
Suppose Yı, Y2, ..., Yn|7 vid N(10, 7-2). The population mean Mo is known. The un-...
Suppose Yı, Y2, ..., Yn|7 vid N(10, 7-2). The population mean Mo is known. The un- known parameter T > 0, which is the inverse of the population variance, is called the precision. The pdf of N(Mo, T-1) is given by Syl-(wl=) = Vb exp (-5(v – wo)"] Let's now derive the posterior distribution of t from the Bayesian perspective. (a) Define U = (Y; – Mo)? i=1 Show that U is a sufficient statistic for t using the Factorization...
Suppose that r(l) is a band-limited signal with the bandwidth W. Suppose that we sampled this...
Suppose that r(l) is a band-limited signal with the bandwidth W. Suppose that we sampled this signal with the sainpling interval T, to generate the sample sequence 1 TLI suppose that 2n/T is larger than the Nyquist rate 2W Given rn, we reconstructed a conius time signal ( using the zero-order-hold method. In other words, rr(l) n for L E [nT, (n +1)T;). In the last lecture, we derived that where s(), as usual, denotes the continuous time representation of...
Suppose we are given two sorted arrays (nondecreasing from index 1 to index n) X[1] ·...
Suppose we are given two sorted arrays (nondecreasing from index 1 to index n) X[1] · · · X[n] and Y [1] · · · Y [n] of integers. For simplicity, assume that n is a power of 2. Problem is to design an algorithm that determines if there is a number p in X and a number q in Y such that p + q is zero. If such numbers exist, the algorithm returns true; otherwise, it returns false....
2 2. Suppose we are given data on n observations (i, Y),, and we have a...
2
2. Suppose we are given data on n observations (i, Y),, and we have a linear model, so that E(X)-A, + ßiri-Let呙-SXY /SXX and β') = F-β,2 be the least-square estimates given in lecture (a) Show that E(SXY)-ASXX and E (T)-A] + β,7. (b) Use (a) to show that E(角)-βι and E(A) = 3). In other words, these are unbiased estimators. (c) The fitted values Yt = Atari are used as estimates of E(A), and the residuals e.-Yi for...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...
12 marks Let independent random samples of sizes n and n2 be taken respectively from two normal distributions with unkn...
12 marks Let independent random samples of sizes n and n2 be taken respectively from two normal distributions with unknown means 1 and 2 and unknown variances oand o. Denote the two samples by . . ,Jn, and y,... , yn2: Which have means T and T, and sample variances s and s2, respectively (a) 4 marks Show that when of = o2, the likelihood ratio test statistic for testing Ho 12 against H 2 can be written as T2...
Suppose two treatments will be given to n patients, randomly sampled from a population. Let uet 1-{ 2 otheatme t one-topat enti, 1 if treatment one is given to patient i 2 otherwise. Let the response be r-{ 1 if treatment t cure patient i, i0 otherwise Here thechance of patient i being given Ti = 1 is 0.5. We want to estimate ㎡ = P07 = 1) for t = 1,2. Assume that (Yin, Ti); i-1 ,n, are independent...
Please help!
(a) Let us consider a full model of a balanced (all t treatments have equal number of observations r) CRD design with t treatments and r replications of each treatment, hence having n-rt observations. 2. i. Minimizing sum of square error Δfull (μ'Ti) -Σι-1 Σ-1 (Vi,- μ-Ti)2 with respect to μ and Ti find the least square estimators of μ and Ti as μ and Ti. Hint: Take derivative of the objective function with respect to μ and...
2. (a) Let us consider a full model of a balanced (all t treatments have equal number of observations r) CRD design with t treatments and r replications of each treatment, hence having n-rt observations i. Minimizing sum of square error Δfull(μ, Tỉ)-Σι-12jai (Vij-l-ri)2 with respect to μ and Ti find the least square estimators of μ and Te as μ and Ti Hint: Take derivative of the objective function with respect to u and Ti and equate then to...
1. Suppose we are going to sample n individuals and ask each sampled person whether they support policy A or not. Let Yi Y0 otherwise 1 if person i in the sample supports the policy, and (a) Assume Y1, , Yn are, conditional on θ. 1.1.d. binary random variables with expec- tation θ. Write down the joint distribution Pr(Yi-yi, . . . ,Ý,-yn(9) in a compact form. Also write down the form of Pr(> Ý,-y|0) (b) For the moment, suppose...
Suppose Yı, Y2, ..., Yn|7 vid N(10, 7-2). The population mean Mo is known. The un- known parameter T > 0, which is the inverse of the population variance, is called the precision. The pdf of N(Mo, T-1) is given by Syl-(wl=) = Vb exp (-5(v – wo)"] Let's now derive the posterior distribution of t from the Bayesian perspective. (a) Define U = (Y; – Mo)? i=1 Show that U is a sufficient statistic for t using the Factorization...
Suppose that r(l) is a band-limited signal with the bandwidth W. Suppose that we sampled this signal with the sainpling interval T, to generate the sample sequence 1 TLI suppose that 2n/T is larger than the Nyquist rate 2W Given rn, we reconstructed a conius time signal ( using the zero-order-hold method. In other words, rr(l) n for L E [nT, (n +1)T;). In the last lecture, we derived that where s(), as usual, denotes the continuous time representation of...
2
2. Suppose we are given data on n observations (i, Y),, and we have a linear model, so that E(X)-A, + ßiri-Let呙-SXY /SXX and β') = F-β,2 be the least-square estimates given in lecture (a) Show that E(SXY)-ASXX and E (T)-A] + β,7. (b) Use (a) to show that E(角)-βι and E(A) = 3). In other words, these are unbiased estimators. (c) The fitted values Yt = Atari are used as estimates of E(A), and the residuals e.-Yi for...
2. Let X1,.n be a random sample from the density 0 otherwise Suppose n = 2m+ 1 for some integer m. Let Y be the sample median and Z = max(Xi) be the sample maximum (a) Apply the usual formula for the density of an order statistic to show the density of Y is (b) Note that a beta random variable X has density f(x) = TaT(可 with mean μ = α/(a + β) and variance σ2 = αβ/((a +s+...
12 marks Let independent random samples of sizes n and n2 be taken respectively from two normal distributions with unknown means 1 and 2 and unknown variances oand o. Denote the two samples by . . ,Jn, and y,... , yn2: Which have means T and T, and sample variances s and s2, respectively (a) 4 marks Show that when of = o2, the likelihood ratio test statistic for testing Ho 12 against H 2 can be written as T2...
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