5. Show that for a normal distribution with mean u and variance o', the central moments satisfy the relation. ika =0 and u = (2K -1)ux-2ơ². !!
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![5. Show that for a normal distribution with mean µ
2
and variance o', the central moments satisfy the
relation.
Hak+1 = 0 and pu, = (2K-1)u,².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc3b3c65-3757-4315-9f2e-c69ef5541fa0%2F4f8d6d84-c101-4825-b84d-2349a650a911%2F5dnivz_processed.jpeg&w=3840&q=75)
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- For mated pairs of gallinules (type of water bird), let X equal the weight in grams of the male and Y the weight in grams of the female. Assume that X and Y have a bivariate normal distribution with #x = 415, o = 611, y = 347 and o=689. The correlation coefficient between X and Y is p = -0.25. (b) Determine the marginal distributions of X and then of Y. (c) Find the conditional distribution of Y|X = x. (d) Find E(Y|X = z) and Var(Y|X = x).A professor grades on a curve by assigning C's to all scores from u- ; to u+%, B's to all scores from i+% to µ+ , and D's to all scores from u – * to µ – 5. Everyone who scores below a D gets an F, and everyone who scores above a B gets an A. a. If the scores are normally distributed with mean u and variance o², what percentage of students will receive each grade? Hint: convert to z-scores and use pnorm() to find probabilities (percentages) b. If the scores are uniformly distributed (continuous) from 0 to 100 what percentage will receive each grade? Hint: first determine the values of u and o for this distributionSuppose X is a random variable of a normal distribution with E(X)=2 and Var(X)=4. Find P(X<4)
- 11. Suppose random variable X is distributed as normal with mean 2 and standard deviation 3 and random variable Y with mean 0 and standard deviation 4, what is the probability density function (pdf) of X + Y.Suppose that f (x) = 0.125x for 0 < x < 4 Determine the mean and the variance of X.A persons blood glucose level and diabetes are closely related XB a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Supposed to after a 12 hour fast the random variable X will have a distribution that is approximately normal with mean of u=87 and a standard deviation of O=25. Note: after 50 years of age both the mean and Standard deviation tends to increase for an adult under 50 after 12 hour fast find the following probabilities A) X is more than 60 B) X is less than 60 C) X is between 60 and 110 D) X is greater than 125 (borderline diabetes starts at 125)
- Use the geometric distribution to derive E(Y), E(Y^2), and V(Y) from the Poisson distributionIf X is a negative binomial rv, then Y= r+Xis the total number of trials necessary to obtainr S’s. Obtain the mgf of Y and then its mean valueand variance. Are the mean and variance intuitively consistent with the expressions for E(X)and V(X)? ExplainThe average height X and weight Y of males in population have a bivariate normal distribution with means µX 1.80 m., µy 90.0 kgs and standard deviations Ox = 0.30 m., oy = 15.3 kgs respectively. The correlation coefficient between X and Y is p= 0.80.
- Let L be the random variable for the length of time, in years, that a person will remember an actuarial statistic. For a certain popula- tion, L is exponentially distributed with mean 1/Y, where Y has a gamma distribution with a = 4.5 and 3 = 4. Find the variance of L.Find P(11<X<15) If X is normally distributed with a mean of 14 and a standard deviation of 2?Let x be a random variable that represents red blood cell count (RBC) in millions of cells per cubic millimeter of whole blood. Then x has a distribution that is approximately normal, For the population of healthy female adults, suppose the mean of the x distribution is about 4.64, Suppose that a female patient has taken six laboratory blod tests over the past several months and that the RBC count data sent to the patient's doctor are as follows. 4.9 4.2 4.5 4.1 4.4 4.3 () Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.) C) Do the given data indicate that the pooulation mean RBC count for this patient is lower than 4.647 Use a- 0.05. (a) What is the level of significance? State the nul and alternate hypotheses. O Hn: 4- 4.64; H: 4.64 O Ho: H> 4.64; H:u- 4.64 O Hn: H- 4.64; H: H 4.64 (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. O The standard normal,…
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