1. Question 1. Properties of response variable in a log

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Question 1
i) Properties of response variable in a logistic model
(a) It is a Bernoulli distribution
(b) It can be any distribution over the interval (0,1)
ii) The logistic function
iii) False
Question 2
i) False
ii) False
iii) Linear regression model
a. False
b. True
c. False
iv) X ^Bis unbiased estimator of XB
v) The e stimate ^δ2=¿∨Y − X ^β||2
2 of δ2 satisfies ( n− p ) ^δ2
δ2 Xn− p
2
vi) Regressionn function f ( x ) of Y =a ¿
Question 3
i) Jeffery’s prior, π j ( λ )=2
ii) Posterior distribution, $ $ π ( I λ X1 ,…, X10 ) =¿ Θ1−α ={θ : p(θ∨Y )≥ hα }
iii) Mean of the posterior distribution, λ⏞
Bayes
=1−α

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iv) Maximum-a-posterior (Map) estimator, λ⏞
MAP
=βnexp(−∑ iyiβ )
v) a=3.222, b=0.978
vi) Comparison of λ⏞
MAP
and λ⏞
Bayes
λ⏞
Bayes
> λ⏞
MAP
because of posterior distribution is skewed to the left
vii) Neither π1 λ nor πJ λ is proper
viii) R is a unique Bayesian confidence region of level 5%
ix) Yes, because our posterior distribution is symmetric and we chose a and b such that (−∞, a )
and ( b , ∞ )have an equal 5% Probability.
x) λ⏞
MLE
=1.2 , λ⏞
MM
=1.44 , a=4.231 , b=6.012
xi) p−value=0.008 ,∧0.242 for LTR
xii) Only Wald’s test rejects the null hypothesis
xiii) The following estimations are all well – defined in the small case: MLE, Method of moments,
Bayes’ estimator, MAP estimator
xiv) True, as the mean of multiple I.I.D exponential random variables is necessarily a Gaussian
distribution
xv) Chi- squared Test
xvi) If he uses the chi – squared test with bins [ 0,1 ) , [ 1,2 ) , … , [ N −1 , N ) , [ N , ∞ )for some , then
the two distributions are indistinguishable.
xvii) It will be possible to tell the distributions apart using a QQ- plots given a sufficiently large
number of samples.
Question 4

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i) MLE of ^P is
∑
i=1
i=n
xi
n
ii) Yes
iii) ^p ( 1− ^p ) is a biasestimator of pq
iv) C=n ¿
v) Delta Method alongwith theCentral Limit Theorem
vi) np
vii) A=np ( 1− p )
Question 5
i) Yes
ii) E ( x ) =z ,Var ( x ) =λ5
iii) λ⏞
MLE
= 1
n ∑
i=1
n
xi
2
iv) λ⏞
MM
= E ( x )
v) Cα ,n= ˇxn +5
Question 6
i) True
ii) We are trying to estimate the magnitude and phase changes undergone by x(in the presence of
vector Gaussian noise)
iii) ∅ =
∑
i−1
n
xi
n
iv) 2 cos x sin x

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v) ~∅ n , MLE=∑
i=1
n
¿ ¿
vi) The MLE of θdoes not change if the covariance matrix of the multivariate Gaussian is scaled
s>0