Let the random variables X and Y have a joint normal distribution with E(X) ar(X) = Var(Y) = 1, and correlation p = -0.6. (a) The random variable Z = 2X + 3Y also has the normal distribution. Find E(Z) and E(Y) = 0, Vor( 7)

Show that variance o? = (x²) – ((x))²

A researcher that wanted to estimate the expectation AY of a random variable Y got three independent observations, Y, Y, Y The researcher knows the value o, of the variance of Y and is considering the following estimators: Pi = 4) Yi + (+) ¥ Py = () Yn + (;) ¥½ + () Y½ in (}) Yi + (}) ¥z + (() %D Which of the following is correct? ONone of the above Ois an unbiased estimator of l and it has the smallest variance of the three estimators. Ois an unbiased estimator of µ and it has the smallest variance of the three estimators. O and i, are both unbiased estimators of fl and Var (ſîz) < Var (îì3). is an unbiased estimator of µ and it has the smallest variance of the three estimators.

Let X1, X2, X3, .., X9 be independent normal random variables with mean ux = 3 and standard deviation ox = 2. Let X be the distribution of the mean of these 9 random variables, namely X = X1+X2++X9 9. (a) What is the shape of the distribution of X? (b) What is the mean of the distribution of X? (c) What is the standard deviation of the distribution of X? Hint: Slide 37, Tutorial 2 (d) Can we determine P(X < 2.5) using a z-score? You do not need to compute this probability, just answer yes or no and briefly explain why or why not. Hint: Slide 26, Tutorial 2

Let X1, X2, X3, .., X36 be independent skewed random variables with mean ux = 3 and standard 2. Let X be the distribution of the mean of these 36 random variables, namely deviation ox X = X1+X2++X36 36 (a) What is the approximate shape of the distribution of X? (b) What is the mean of the distribution of X? (c) What is the standard deviation of the distribution of X? (d) Can we determine P(X < 2.5) using a z-score? You do not need to compute this probability, just answer yes or and briefly explain why or why not.

Two airplanes are flying in the same direction in adjacent parallel corridors. At time t = 0, the first airplane is 10 km ahead of the second one. Suppose the speed of the first plane (km/hr) is normally distributed with mean 545 and standard deviation 9 and the second plane's speed is also normally distributed with mean and standard deviation 525 and 9, respectively.   (a) What is the probability that after 2 hr of flying, the second plane has not caught up to the first plane? (Round your answer to four decimal places.)(b) Determine the probability that the planes are separated by at most 10 km after 2 hr. (Round your answer to four decimal places.)

If two independent sufficiently large samples are taken from two populations with known population standard deviations, the sampling distribution of the difference, sample mean (/x)1 - sample mean (/x)2 , between the two sample means /x1 and /x2 a. can be approximated by poisson distribution b. will have a variance of one. c. can be approximated by a normal distribution d. will have a mean of one.  I would like to know why the answer is the answer and thank you.

The sales manager of a large company selected a random sample of n = 10 salespeople and determined for each one the values of x = years of sales experience and y = annual sales (in thousands of dollars). A scatterplot of the resulting (x, y)pairs showed a linear pattern.(a)Suppose that the sample correlation coefficient is r = 0.75 and that the average annual sales is y = 100.If a particular salesperson is 2 standard deviations above the mean in terms of experience, what would you predict for that person's annual sales?Since the value for experience is 2 standard deviations above the mean experience, you can predict annual sales will be ________________standard deviations above the mean annual sales.(b)If a particular person whose sales experience is 1.7 standard deviations below the average experience is predicted to have an annual sales value that is 1 standard deviation below the average annual sales, what is the value of r? (Round your answer to two decimal places.)r =

* Let X is a Gaussian random variable with zero mean an variance o. Y = X. Find mean of Random Variable Y.

4. The battery life of a fully charge iPhone 13 Pro on a standby follows a normally distributed with mean 7 hour and variance o² fl = 1 hours. = (a) Write out the integral form of the probability that a fully-charged iPhone 13 Pro last between 7 and 9 hours on standby. then use the pnorm function in R to find this probability. (b) What is the probability that a fully-charged iPhone 13 Pro last less than 5 hours? (c) What is the probability that a fully-charged iPhone 13 Pro last more than 9 hours? (d) What is the minimum time a fully-charged iPhone 13 Pro need to last in order to be in the top 10% best quality?