Question:

irquality is a built-in R data frame. The airquality data frame is a time series giving the daily air quality measurements in New York, May to September 1973. We are interested in some descriptive statistics related to the Wind column of the airquality data framet. We can access the data directly by using the assignment x

a.Calculate the sample mean of x.

b. Using the R quantile function, find the .25 (25th) quantile of x.

c. Calculate the interquartile range of x using R.

d. Calculate the sample median of x.

e. Calculate a 4/153 trimmed mean for x.

f. Calculate the sample variance of x.

g. Calculate the sample standard deviation of x.

h. What proportion of the x values are within 1.0 sample standard deviations from the sample mean

i. Calculate the maximum of x.

j. What number do you get if you add up the square roots of the values in x?

Question# 2: Complete the following sentences: 1) If the random process X(t) represents the total number of events occurred in the interval [0, t) then, the X(1) is called a 2) The Interarrival times for the Poisson process are identical independent distributed. 3) Poisson Process has an --------- and increments . 4) X(1) - X(s) in a Poisson Process represent that the number of events occurred in the interval is 5 events. 5) Poisson process {X(t)It 2 0) is a counting process which X() are ------- and -------- Values.Consider a Markov chain {Xn : n = 0, 1, 2, ...} with state space {1, 2, 3} and one-step transition probability matrix O NIH NIH P = O 0 O (a) Mark O or X: ( ) The Markov chain is irreducible. ( ) The Markov chain is aperiodic. ( ) The Markov chain is transient. ( ) The Markov chain is recurrent. ( ) The Markov chain is null recurrent. ( ) The Markov chain is ergodic. (b) Calculate P(X5 = 1/X2 = 1). (c) Find limn + P(Xn = 1/X2 = 1).Question# 2: Complete the following sentences: 1) If the random process X(t) represents the total number of events occurred in the interval [0, t) then, the X(7) is called a ---....--=>. 2) The Interarrival times for the Poisson process are identical independent --------- distributed. 3) Poisson Process has an ---.----- and increments. 4) X(1) - X(s) in a Poisson Process represent that the number of events occurred in the interval mmm------ is 5 events. 5) Poisson process {X(t) It 2 0} is a counting process which X(t) are ------- and values.Problem 2. The eigenvalues of the following matrix are 5,0, -1. Find the associated eigenvectors manually, and then use those to find an orthogonal diagonalization. A = 0 0 You may use Matlab or a calculator to do any difficult computations, but do not use it to find the eigenvectors or orthogonal diagonalization directly. (b) Use you answer from part (a) to find the spectral decomposition of A. You may use Matlab to do the spectral decomposition computations, but not to find the spectral decom- position directly. Write your answer in the form described in Problem (1b), except now it will be in the form AM, + AM2 + AsMa for some M1, M2, and MS