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Problem 4: Consider the problem of estimating the unknown parameter p of a Bernoulli random varia...
50] 1. Suppose that Xi,X2.. are independent and identically distributed Bernoulli random vari-abl...
50] 1. Suppose that Xi,X2.. are independent and identically distributed Bernoulli random vari-ables with success probability equal to an unknown parameter p E (0, 1). Let P,-n-1 Σǐl Xi denote the sample proportion. liol a. Ti, what des VatRtA-P) converge in law ? 10 a. To what does)converge in law ? [10] b. Use your answer to part a to propose an approximate 95% confidence interval for p. 10 c. Find a real-valued function g such that vn(g(p) -g(p)) converges...
Problem:2 Let Xi, X2,... , Xn be a random sample from Bernoulli(p) and consider es- timators...
Problem:2 Let Xi, X2,... , Xn be a random sample from Bernoulli(p) and consider es- timators iand p2- i. Compute the mean square error (MSEp) for both estimators p? and p2 Note that you must show the details of the calculation to receive full credit. ii. Use R to plot MSE, for both estimators using sample sizes n 20 and n- 300. Comment on the plots. iii. Use R to simulate 10,000 different Bernoulli samples of n 300 with success...
Problem 6 7. Part 1-4 (10pts)A coin has two faces Head and Tail. (1) (2pts)]lf you...
Problem 6
7. Part 1-4
(10pts)A coin has two faces Head and Tail. (1) (2pts)]lf you toss the coin once, and record the up-face value, what is the sample space? 6. (2) (2pts)lf you toss the coin once, what is the probability that up-face is Tail? What is the probability that up-face is Head? (3) (5ps)lf you toss the coin three times, and record the up-face value for each toss. One of the possible outcome is (Head, Head, Head). By...
Problem 4, 5 p. ] (in prepation to the binomial model) Consider tossing a coin n...
Problem 4, 5 p. ] (in prepation to the binomial model) Consider tossing a coin n times where n 1 is fixed. Assume that the probability of occurring of "heads" is p(0< p1), and the probability of occurring of "tails" is q1-p and the outcomes of single tosses are independent of each other. Describe the sample space Ω of that experiment (all possible outcomes) and how the corresponding probability function P on Ω looks like. In other words, prescribe P...
Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses...
Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses of a biased coin, at times k = 0,1,2,...,n On each toss, the probability of Heads is p, and the probability of Tails is 1 -p {1,2,.., at time for E resulted in Tails and the toss at time - 1 resulted in A reward of one unit is given if the toss at time Heads. Otherwise, no reward is given at time Let...
Concept Check: Terminology 0/3 points (graded) Suppose you observe iid samples X1,…,Xn∼P from some unknown distribution...
Concept Check: Terminology
0/3 points (graded)
Suppose you observe iid samples X1,…,Xn∼P from some
unknown distribution P. Let F denote a parametric
family of probability distributions (for example, F could be the
family of normal distributions {N(μ,σ2)}μ∈R,σ2>0).
In the topic of goodness of fit testing, our
goal is to answer the question "Does P
belong to the family F, or is P
any distribution outside of F
?"
Parametric hypothesis testing is a particular case of goodness
of fit testing...
Consider the problem of estimating π using a unit square centered at (1/2, 1/2) and an...
Consider the problem of estimating π using a unit square
centered at (1/2, 1/2) and an inscribed circle inside the square.
We will estimate π by simulating n darts. For the nth dart, if the
dart is inside the circle, then we return In = 1; otherwise, we
return In = 0.
1. Are I1, I2, · · · , In independent? Under what
assumption?
2. Are I1, I2, · · · , In identically distributed?
3. Let p represent...
50] 1. Suppose that Xi,X2.. are independent and identically distributed Bernoulli random vari-ables with success probability equal to an unknown parameter p E (0, 1). Let P,-n-1 Σǐl Xi denote the sample proportion. liol a. Ti, what des VatRtA-P) converge in law ? 10 a. To what does)converge in law ? [10] b. Use your answer to part a to propose an approximate 95% confidence interval for p. 10 c. Find a real-valued function g such that vn(g(p) -g(p)) converges...
Problem:2 Let Xi, X2,... , Xn be a random sample from Bernoulli(p) and consider es- timators iand p2- i. Compute the mean square error (MSEp) for both estimators p? and p2 Note that you must show the details of the calculation to receive full credit. ii. Use R to plot MSE, for both estimators using sample sizes n 20 and n- 300. Comment on the plots. iii. Use R to simulate 10,000 different Bernoulli samples of n 300 with success...
Problem 6
7. Part 1-4
(10pts)A coin has two faces Head and Tail. (1) (2pts)]lf you toss the coin once, and record the up-face value, what is the sample space? 6. (2) (2pts)lf you toss the coin once, what is the probability that up-face is Tail? What is the probability that up-face is Head? (3) (5ps)lf you toss the coin three times, and record the up-face value for each toss. One of the possible outcome is (Head, Head, Head). By...
Problem 4, 5 p. ] (in prepation to the binomial model) Consider tossing a coin n times where n 1 is fixed. Assume that the probability of occurring of "heads" is p(0< p1), and the probability of occurring of "tails" is q1-p and the outcomes of single tosses are independent of each other. Describe the sample space Ω of that experiment (all possible outcomes) and how the corresponding probability function P on Ω looks like. In other words, prescribe P...
Problem 5. Indicator variables S points possible (graded) Consider a sequence of n 1 independent tosses of a biased coin, at times k = 0,1,2,...,n On each toss, the probability of Heads is p, and the probability of Tails is 1 -p {1,2,.., at time for E resulted in Tails and the toss at time - 1 resulted in A reward of one unit is given if the toss at time Heads. Otherwise, no reward is given at time Let...
Concept Check: Terminology
0/3 points (graded)
Suppose you observe iid samples X1,…,Xn∼P from some
unknown distribution P. Let F denote a parametric
family of probability distributions (for example, F could be the
family of normal distributions {N(μ,σ2)}μ∈R,σ2>0).
In the topic of goodness of fit testing, our
goal is to answer the question "Does P
belong to the family F, or is P
any distribution outside of F
?"
Parametric hypothesis testing is a particular case of goodness
of fit testing...
Consider the problem of estimating π using a unit square
centered at (1/2, 1/2) and an inscribed circle inside the square.
We will estimate π by simulating n darts. For the nth dart, if the
dart is inside the circle, then we return In = 1; otherwise, we
return In = 0.
1. Are I1, I2, · · · , In independent? Under what
assumption?
2. Are I1, I2, · · · , In identically distributed?
3. Let p represent...
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