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.

A student has to do 100 homework problems. The time taken for each of the problems is a uniform random variable with a mean of 4 minutes and a variance of 2.25 = 1.5^2 . Assume that the time taken for the problems are independent. Approximate the probability that the student does not finish her homework in 420 minutes.

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 X1 ∼ N(μ = 2, σ^2 = 4), X2 ∼ N(μ = 1, σ^2 = 16), and X3 ∼N(μ = −4, σ^2 = 9). Let X1, X2, and X3 be independent and Y = (X1 + 2X2 + X3)^2. What is the mean and variance 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}

Let x be a random n-vector with mean mu and positive definite variancecovariance matrix Sigma. Define Sigma^(-(1)/(2))=(Sigma^((1)/(2)))^(-1). (a) Show that Sigma^(-(1)/(2))(x-mu) has mean 0 and variance-covariance matrix I. (b) Determine E[(x-mu)^(T)Sigma^(-1)(x-mu)] and var[(x-mu)^(T)Sigma^(-1)(x-mu)], assuming for the latter that the skewness matrix Lambda and excess kurtosis matrix Omega of x exist.

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?