1. Draw a pdf (or histogram) and its cdf of the sum of 5 iid uniform...

1. Draw a pdf (or histogram) and its cdf of the sum of 5 iid uniform random variables and compare with Gaussian pdf that can be obtained from CLT.

2. Do the same thing with 50 iid uniform random variables.

Expert Solution

orchestra answered 1 year ago

Suppose that T (≥ 0) is a continuous random variable. Let its pdf, cdf, survival function,...

Suppose that T (≥ 0) is a continuous random variable. Let its pdf, cdf, survival function, hazard rate function, and cumulative hazard rate function be f(t), F(t), s(t), h(t), and H(t), respectively. Note that H(t) is defined by R t 0 h(u)du. a. Denote s(t), h(t), and H(t) as a function of f(t). b. Denote f(t), h(t), and H(t) as a function of s(t). c. Denote f(t), s(t), and H(t) as a function of h(t). d. Denote f(t) and s(t)...

Let ?1, ?2,…. . , ?? (n random variables iid) as a variable whose pdf is...

Let ?1, ?2,…. . , ?? (n random variables iid) as a variable whose pdf is continuous and uniform over the interval [? - 1; ? + 3]. (1) Determine the estimator of the moments method. (2) Is this estimator unbiased? What is its variance? (3) Find the maximum likelihood estimator (VME) for this setting. Is it unique?

Let X1,…, Xn be a sample of iid random variables with pdf f (x; ?1, ?2)...

Let X1,…, Xn be a sample of iid random variables with pdf f (x; ?1, ?2) = ?1 e^(−?1(x−?2)) with S = [?2, ∞) and Θ = ℝ+ × ℝ. Determine a) L(?1, ?2). b) the MLE of ?⃗ = (?1, ?2). c) E(? ̂ 2).

Let X1,…, Xn be a sample of iid Exp(?1, ?2) random variables with common pdf f...

Let X1,…, Xn be a sample of iid Exp(?1, ?2) random variables with common pdf f (x; ?1, ?2) = 1/ ?1 e^ −( x−?2)/ ?1 for x > ?2 and Θ = ℝ × ℝ+. a) Show that S = (X(1), ∑n i=1 Xi ) is jointly sufficient for (?1, ?2). b) Determine the pdf of X(1). c) Determine E[X(1)]. d) Determine E[X^2 (1) ]. e) Determine Var[X(1)]. f ) Is X(1) an MSE-consistent estimator of ?2? g) Given...

3. [5+5+1=11 pts] a. Draw a frequency histogram for “Temp” variable from “airquality” data. Use breaks=seq(55,...

3. [5+5+1=11 pts] a. Draw a frequency histogram for “Temp” variable from “airquality” data. Use breaks=seq(55, 100, 5) in your hist() function. Show frequency on top of each bar. Paste your graph below. b. Draw a relative frequency histogram for “Temp” variable from “airquality” data using the same ‘breaks’ parameter. Add a density curve to it. Paste your graph below. c. What kind of distribution does the data exhibit? Right skewed? Left skewed? Symmetric?

Let X1…X5 be iid random sample of size n=5 from a Uniform distribution with parameters 0,theta....

Let X1…X5 be iid random sample of size n=5 from a Uniform distribution with parameters 0,theta. We test H0: theta=1 versus H1: theta=2. Reject H0 when max(X1…X5)>b. Find the value of b so that the size of the test is 0.10, then find the power of this test. Among the tests based on max(X1…Xn), find the most powerful test with size exactly equal to 0.

Let X1, ..., Xn be iid with pdf f(x; θ) = (1/ x√ 2π) e(-(logx- theta)^2)...

Let X1, ..., Xn be iid with pdf f(x; θ) = (1/ x√ 2π) e(-(logx- theta)^2) /2 Ix>0 for θ ∈ R. (a) (15 points) Find the MLE of θ. (b) (10 points) If we are testing H0 : θ = 0 vs Ha : θ != 0. Provide a formula for the likelihood ratio test statistic λ(X). (c) (5 points) Denote the MLE as ˆθ. Show that λ(X) is can be written as a decreasing function of | ˆθ|...

Construct a two-sided (1−α)-CI for θ in Uniform[0,θ] based on IID Y1,...,Yn

Let ?1. . . ?5 be identically independently distributed (iid) variables sampled from a binomial distribution...

Let ?1. . . ?5 be identically independently distributed (iid) variables sampled from a binomial distribution Bin(3,p). a) Compute the likelihood function (LF). b) Adopt the appropriate conjugate prior to the parameter p, choosing hyperparameters optionally within the support of distribution. c) Using (a) and (b), find the posterior distribution of p. d) Compute the minimum Bayesian risk estimator of p.

5. Find the sum of terms in given arithmetic sequence 1 + 3 + 5 +...

5. Find the sum of terms in given arithmetic sequence 1 + 3 + 5 + ... + 59 6. Find the sum of terms in given arithmetic sequence 2 + 5 + 8 + ... + 41 7.Given a geometric sequence 6 + 2 + 2/3 + ... is this sequence converging or diverging, if it is converging find it's sum

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