Let X1, . . . , Xn be a random sample from an exponential distributi.pdf

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Let X1, . . . , Xn be a random sample from an exponential distribution with thedensity function f (x|? ) = ? exp[?? x]. Derive a likelihood ratio test of H0: ? =?0 versus HA: ? = ?0, and show that the rejection region is of the form:{X exp[??0 X] ? c}. Solution likelihood ratio test = lambda = L(1/mean(X),X1)*L(1/mean(X),X2)* L(1/mean(X),X3)*...........L(1/mean(X),Xn)/L(?0,X1)*L(?0,X2)* L(?0,X3)*...........L(?0,Xn) = (1/mean(X)) *exp[?1/mean(X)* x1+x2+....xn]/?0 exp[??0* (x1+x2+....xn)] since the m.l.e of theta = 1/mean(X) = (1/mean(X)) *exp[?n]/?0 exp[??0* (x1+x2+....xn)] since?0& exp[?n] are constant , we reject when , (n/summation(Xi))/exp[??0* (x1+x2+....xn)] is more than a critical value i.e. when summation(Xi)*exp[??0* summation(Xi)] < c.

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like many entreneurs, Tony Smith started Sonic as a sole proprietorship, then took on a partner, and eventually offered franchises. what advantages did he enjoy at each stage of Sonic\'s development? What disadvantages did he face? Solution There are a number of disadvantages of the company organization as a sole proprietorship: Advantages sole proprietorship: Disadvantages of changing the company organization to partnership: Advantages partnership: Advantages of changing the company organization to franchises: Disadvantages of changing the company organization to franchises:.

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Lewotontin wrote a critical review of The Social Organization of Sexuality by E.O. Laumann and others. Laumann was using data from a sample survey, in which respondents answer questions about their sexual behavior, including the number of partners in the previous five-year period. On average, among heterosexuals, men reported having about twice as many partners as women. Lewotontin thought this was a serious inconsistency, showing that respondants \"are telling themsevles and other enormous lies\" Laumann replied that you should not use averages to summarize such skewed and long-tailed distributions. a) Why is it inconsistent for men to report having twice as many partners as women? Solution Yes the data is inconsistent, for men to report having twice as many partners as women, because of the survey data is response variable. because of the survey data is response variable..

Lewotontin wrote a critical review of The Social Organization of Sex.pdf

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Let Z_{n} denote the sample mean for a random sample of size n from the standard normal distribution. For n = 1, 4 and 16 1. sketch the probability.distibution.function of each Z_{n} in the same coordinate system, 2. compute the quartiles of each Z_{n} Solution We begin our journey into inferential statistics. Most of the time the population mean and population standard deviation are impossible or too expensive to determine exactly. Two of the major tasks of a statistician is to get an approximation to the mean and analyze how accurate the approximation is. The most common way of accomplishing this task is by using sampling techniques. Out of the entire population the researcher obtains a (hopefully random) sample from the population and uses the sample to make inferences about the population. From the sample the statistician computes several numbers such as the sample size, the sample mean, and the sample standard deviation. The numbers that are computed from the sample are called statistics..

$Let Z_{n} denote the sample mean for a random sample of size n from.pdf$ $Let Z_{n} denote the sample mean for a random sample of size n from.pdf$

Let Z_{n} denote the sample mean for a random sample of size n from.pdf

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LetX_1, X_2, . . . , X_nbe i.i.d. from distributionF: 1. IfFis theUniform(-theta,0) distribution, find the Maximum likelihood explanation oftheta. Show work for credit. 2. IfFis theUniform(?theta,theta) distribution, find the Maximum likelihood estimation oftheta. Show work for credit. Solution Suppose there is a samplex1,x2, ...,xnofnindependent and identically distributedobservations, coming from a distribution with an unknownprobability density functionf0(.

LetX_1, X_2, . . . , X_nbe i.i.d. from distributionF 1. IfFis t.pdf

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Let\'s say you frequently drink wine that\'s been rated highly by Robert Parker, one of the wine experts. His ratings were demonstrated to be reliable and you find that you usually agree with Robert Parker. How does this observation reflect the concept of validity? Solution As you know that validity is \"the degree to which evidence and theory support the interpretations of test scores\". Since the rating of him is reliable to as as you agree his experties so the Concept of validity is True here..

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Let X1,X2,X3 and X4 be independent random variables. Assume that X2,X3 and X4 each are poison distributed with mean 5, suppose Y=X1+X2+X3+X4~ POI(25). What is the distribution of X1. Solution pendent random variables. Assume that X2,X3 and X4 each are poison distributed with mean 5, suppose Y=X1+X2+X3+X4~ POI(25). What is the distribution of X1. Y will be poissone with mean of 5 + 5 + 5 + X1 so 5 + 5 +5 + x1 = 25 ( but given is 25) x1 = 25 - 15 x1 = 10 X1 is poison random with mean = 10.

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Let Y1 be the number of success in n = 10 independent Bernoulli trials with probability of success Î¸. Suppose the four possible values for Î¸ are Î© = {0.2, 0.4, 0.6, 0.8}. We observe Y1 = 7. (a) Find the posterior distribution for Î¸ if we use a uniform prior distribution (i.e. a vague, non-informative prior). (b) Find the posterior distribuiton for Î¸ if the prior distribution is P(Î¸ = 0.2) = 0.1, P(Î¸ = 0.4) = 0.2, P(Î¸ = 0.6) = 0.3, P(Î¸ = 0.8) = 0.4. HINT: You may wish to use R by installing the Bolstad package and utilizing the binodp function. Solution plz repost the question.

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Let Y be a continuous random variable and define m as the median of the distribution (not the sample median). If Y1,..,Y4 are independent random variables from the same continuous distribution, what is the probability that the largest exceeds the median? Solution Y1, Y2 , Y3 ,Y4 Let them be in increasing order then median is the average of middle values so it must be in between y2 and y3 so we have two values greater than median so the probability that the largest exceeds the median = 2/4 = 1/2 or 0.5.

Let Y be a continuous random variable and define m as the median of .pdf

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Let X1 and X2 have a trinomial distribution. Differentiate the moment-generating function to show that their covariance is -np1p2 Solution The moment generating function is (p1e^t1` + p2e^t2 + 1 - p1 - p2)^n Using the derivative with repect to t1 and t2 gives us E[x1x2] Then, var(x1,x2) = E(x1x2) - E(x1)E(x2) We know that E(x1) = np1 and E(x2) = np2. We can also see this from the first derivative of the moment generating function. The derivative with respect to t1 is np1e^t1(p1e^t1` + p2e^t2 + 1 - p1 - p2)^(n-1) (Note that this gives us E(x1) = np1; calculating for x2 similiarly gives us E(x2) = np2) Then, the derivative with respect to t2 is np1e^t1(n-1)p2e^t2(p1e^t1` + p2e^t2 + 1 - p1 - p2)^(n-2) Evaluating this at t1 = t2 = ...=tk-1 = 0, we getnp1(n-1)p2(1)^n-2= np1(n-1)p2 Then the covariance isnp1(n-1)p2 - np1(np2) = p1p2(n^2-n)-n^2p1p2 = p1p2(n^2-n-n^2) = -np1p2.

Let X1 and X2 have a trinomial distribution. Differentiate the momen.pdf

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