Define a 3-dimensional random vector X = (X₁, X₁, X3)'. Also define the following a.) the joint CDF and joint PDF of X (continuous (ar) b.) all marginal (joint) CDFS of X.
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- The 4-dimensional random vector X has PDF fX(x)={1 when 0≤xi≤1, i=1,2,3,4 and 0 otherwise}. Are the for components of X independent random variables?Let X1, X2, ..., Xn ~ exp(A) be a random sample. Show that X is sufficient for 1/A.The built-in data set, UKgas, is a time series recording the quarterly UK gas consumption from 1960 to 1986. Using R we can convert this data into the vector x by the assignment x30 we can create a confidence interval for u using a normal critical value. If we want the confidence interval to be at the 94% level and we use a normal critical value, then what critical value should we use? i) Calculate a 94% confidence interval(using a normal critical value) for μ. j) How long is the 94% confidence interval just created in part i?
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- Suppose that the continuous two-dimensional random variable (X, Y ) is uniformly distributed over the square whose vertices are (1, 0), (0, 1), (−1, 0), and (0, −1). Find the Correlation Coefficient ρxyConsider random variables X1, · · · Xn, independent and identically distributed such that each Xi ∼N (0, 1). Write down an expression for the joint pdf of the n-dimensional random vector (X1, · · · Xn).(i.e. what is the distribution of a random sample of n standard normal random variablesLet X and Y be independent Exp(2) random variables. Define W = X + Y. a) Determine the correlation between X and W. The joint pdf of X and W is: fx,w(x, w) = 1^(2) e^(-1w) I{OSEE MORE QUESTIONS