We have the following information about the random variables X and Y: o² = 4.6, o} = 8.5, and pxy = 0.4. Calculate the variance of Z = 4X + 5Y.

The number of offspring of an organism is a discrete random variable withmean μ and variance σ^2. Each of its offspring reproduces in the same manner. Find the expected number of offspring in the third generation and its variance.

Show that the variance of a geometric random variable X; is given by V ar(X) = q/p^2 .

Let X be a random variable with mean µ and variance σ^2. Let Y = aX2 + bX + c. Find the expected value of Y.

Let X be a normal random variable with mean μ=10 and variance σ^2=36. Find (a)P{X<22} (b)P{X>5} (c)P{4<X<16}

Researches are interested in whether or not the average fuel economy for compact cars is thesame as sedan cars. A sample of 36 compact cars and 25 sedan cars is taken.The sample of compacts returns a mean of ̄X = 38 miles per gallon and sample variance of s^2 = 25.The sample of sedans returns a sample mean ̄Y = 36 with a sample variance of s^2 = 49.Let μ 1 be the population mean of compacts and μ 2 for sedans. a. Create a 95% confidence interval for μ 1 .b. Now create a 95% confidence interval for μ 1 μ 2 .c. Say the interval from part b. contains 0. What can you say about the difference in the averagefuel economy of compact cars versus economy cars

Explain at least two differences between the following two point estimations of the population variance. 22 = Σ'1(X, – X)2 n-1 62 _ Σ 1 (X - X)2 =1 = n

The scores on a placement test given to college freshmen for the past five years are approximately normally distributed with a mean μ = 74 and a variance σ^2 = 8. Would you still consider σ^2 = 8 to be a validvalue of the variance if a random sample of 20 students who take the placement test this year obtain a value ofs^2 = 20?

Suppose that 90% of Narnians are children, that Narnian children have normally distributed BMIs with a mean of 25 kg/m​^2 and variance of 64 (kg/m​^2)^2​ , and that Narnian adults have normally distributed BMIs with a mean of 30 kg/m​^2 and variance of 100 (kg/m​^2)^2​ . WHO currently categorizes BMIs greater than 30 as obese. a. What is the prevalence of obesity among Narnian children? b. What is the prevalence of obesity among Narnian adults? c. What is the overall prevalence of obesity among Narnians? d. Are age and obesity statistically independent in Narnia? e. What proportion of Narnians are obese children? f. What proportion of obese Narnians are children?

Let X1,...,Xn be a sample from normal with mean theta and variance 1. Construct an unbiased estimator of theta^2 based from X^2. Find its variance and compare it with the Cramer-Rao lower bound.