Number...2 - abx x, x² A-ay-A-Font2.3.5FParagraphStylesdLet Y₁, Y2, Y3, Y4, Ys and X1, X2, X, be independent and normally distributed random samplesfrom populations with means μ₁ = 2 and 2 = 8 and variances 01²=5 and 02² = k, respectively.Suppose that P(X-Y> 10) = 0.02275, find the value of 0₂² = k6.NormalSuppose that Y₁, Y2, Ym and X₁, X2, ..., Xm are independent normally distributed random samplesfrom populations with means ₁ and ₂ and variances 01² and 02², respectively. IsX-Y a consistent estimator of 2-1? Justify your answer.| 4. Suppose that Y₁, Y2... Ym is a random sample of size m from Gamma(a = 3, p = 0), where is notknown. Check whether or not the maximum likelihood estimator is a minimum variance unbiasedestimator of the parameter 0.WORDS EX ENGLISH (SOUTH AFRICA)- here to search1 No Spac... Heading 1 Heading 2TitleSuppose that a random sample X₁, X2, X₂0 follows an exponential distribution with parameter pCheck whether or not a pivotal quantity exixts, if it exists, find a 100(1 - a)%Dconfidence interval for p.Suppose that a random sample X is given by a probability density function(B-2), 0

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…
Let X1, X2 be two independent random variables with the same mean EXi = µ andpossibly different variances Var(Xi) = σ2i (sigma squared i), i = 1, 2. Consider the weighted average Y =λX1 + (1 − λ)X2 where λ is a constant.(a) Compute EY and Var(Y )(b) Find the λ in terms of σ2i (sigma squared i) , i = 1, 2 that minimizes Var(Y ).
If you let X1, X2, X3, X4 equal the cholesterol level of a woman under the age of 50, a man under 50, a woman 50 or older, and a man 50 or older, respectively. Assuming the distribution of Xi is N(μi, σ2), i = 1, 2, 3, 4 and you test the null hypothesis H0: μ1 = μ2 = μ3 = μ4, using seven observations of each Xi, what would be the critical region for an alpha = 0.05 significance level?
A snack food manufacturer estimates that the variance of the number of grams of carbohydrates in servings of its tortilla chips is 1.33. A dietician is asked to test this claim and finds that a random sample of 24 servings has a variance of 1.37. At α=0.01, is there enough evidence to reject the manufacturer's claim? Assume the population is normally distributed. Complete parts (a) through (e) below. (a) Write the claim mathematically and identify H0 and Ha. A. H0: σ2≤1.33 (Claim) Ha: σ2>1.33 B. H0: σ2≠1.33 Ha: σ2=1.33 (Claim) C. H0: σ2≥1.33 Ha: σ2<1.33 (Claim) D. H0: σ2=1.33 (Claim) Ha: σ2≠1.33 (b) Find the critical value(s) and identify the rejection region(s). The critical value(s) is(are) enter your response here. (Round to two decimal places as needed. Use a comma to separate answers as needed.) Choose the correct statement below and fill in the corresponding answer boxes. A. The…
A random sample of n1=17�1=17 securities in Economy A produced mean returns of x̄ 1=5.6% x̄ 1=5.6% with s1=2.3%�1=2.3% while another random sample of n2=20�2=20 securities in Economy B produced mean returns of x̄ 2=4.6% x̄ 2=4.6% with s2=2.3%.�2=2.3%. At α =0.2 α =0.2, can we infer that the returns differ significantly between the two economies? Assume that the samples are independent and randomly selected from normal populations with equal population variances ( σ 12= σ 22)( σ 12= σ 22). T-Distribution Table a. Calculate the test statistic. t=�= Round to three decimal places if necessary b. Determine the critical value(s) for the hypothesis test. + Round to three decimal places if necessary c. Conclude whether to reject the null hypothesis or not based on the test statistic. Reject Fail to Reject
A chemical supply company currently has in stock 100 lb of a certain chemical, which it sells to customers in 4-lb batches. Let X = the number of batches ordered by a randomly chosen customer, and suppose that X has the following pmf. x 1 2 3 4 p(x) 0.3 0.5 0.1 0.1 Compute E(X) and V(X). E(X) = batches V(X) = batches2 Compute the expected number of pounds left after the next customer's order is shipped and the variance of the number of pounds left. [Hint: The number of pounds left is a linear function of X.] expected weight left lb variance of weight left lb2
Suppose a linear model y=β0+β1xy=β0+β1x is fit to a sample data set, and a test of the null hypothesis H0:β1=0H0:β1=0 against an alternative hypothesis HA:β1≠0HA:β1≠0 is performed; a PP-value of 0.4203 is obtained. Which of the following scatter plots depicts the data set on which this model was fit and the hypothesis test was performed?
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Suppose that Y1,Y2,Y3 denote a random sample from an exponential distribution with density function f(y) = Consider the following four estimators of θ: ?1θe−y/θ, y>0,0, otherwise. θˆ =Y, θˆ =Y1+Y2, θˆ =Y1+2Y2, θˆ =Y1+Y2+Y3 =Y ̄. 11223343 Which estimators are unbiased? Among the unbiased estimators, which has the smallest variance?
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The random variable X has a Bernoulli distribution with parameter p. A random sampleX1, X2, . . . , Xn of size n is taken of X. Show that the sample proportionX1 + X2 + · · · + Xnnis a minimum variance unbiased estimator of p.Given that X1, X2, . . . , Xn forms a random sample of size n from a geometric population withparameter p, show thatY =n∑j=1

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2. Title 5 Subtitle Paragraph Styles Let Y₁, Y2, Y3, Y4, Ys and X₁, X2, X, be independent and normally distributed random samples from populations with means ₁ = 2 and 2 = 8 and variances 0₁²=5 and 02²= k, respectively. Suppose that P(X-Y> 10) = 0.02275, find the value of 0₂² = k www