We have a random variable X and Y that have the joint pdf f(x) = {1    0

Let X be a random variable and a real number. Show that E(X - a)² = varX + (µ − a)² Hereμ = EX is the expected value of the random variable X and varX = E(X - μ)^2 is the variance of the random variable X. Guidance: start from the representation - (X-a)^2 = (X µ + μ- a)^2 and group the right side of the representation appropriately into the form (Z + b)^2, where Z is some random variable and b is a real number and open the square. The task should be solved with the help of the expected value calculation rules.

The random variables X,Y have variance Var(X)=36 and Var(Y)=1 and their correlation is Cor(X,Y)=−3/4. Calculate Var(X+Y) with a full explanation

Suppose that X and Y are random variables where the conditional expected value of Y given X is given by E (Y|X) = 3X + 6 If X has the expected value 0 and variance Var(X)=9 determine the minimal possible value of the variance Var(Y) of Y./

Let ZZ be a discrete random variable taking one of the four distributions covered in Chapter 10. Suppose you know that Var(Z)=(k+1)E(Z)Var(Z)=(k+1)E(Z) where kk is the last non-zero digit of your student ID number. Determine the distribution of ZZ and find its parameter(s), explaining your argument carefully.

Let X1 be a normal random variable with mean 2 and variance 3 and let X2 be a normal random variable with mean 1 and variance 4. Assume that X1 and X2 are independent. (a) What is the distribution of the linear combination Y = 2X1 + 3X2? (b) What is the distribution of the linear combination Y = X1 – X2?

Inspection trials are carried out in a factory to assess the quality of products. In each trial r¡ products are tested and Y;, the number of satisfactory items in the trial, is assumed to follow the binomial model Y; Bin(ri, p), where p is the proportion of satisfactory items in the production process. Using inspection data (rı, yı), (r2, Y2), …‚ (˜n, Yn) and assuming independent trials, the following log-likelihood for p was obtained ... l(p; data) = 25.8392 + 35 · log(p) + 17 · log(1 − p). A) Determine which of the following (r1, y₁), (r2, y2), …‚ (˜n, yn) is the correct data that yields the given log-likelihood 1(p; data). Note that you do not need to develop the likelihood from scratch, but to carefully look and use results shown in lectures, notes and exercises. B) Write the maximum likelihood estimate p.

Show that variance o? = (x²) – ((x))²

An ordinary (fair) coin is tossed 3 times. Outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is hht, then R (hht)=1. Suppose that the random variable X is defined in terms of R as follows X=6R-2R^2-1. The values of X are given in the table below. A) Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values X. Then fill in the appropriate probability in the second row.

If a random process, X(t)= Acos wt + B sin wt is given, where A and B are uncorrelated, zero mean random variables having the variance o, find (a) autocorrelation