Problem 1. Consider the two data collections (x,..-, X) and (y,. ya). The median difference statistic...
Problem 1. Let X and Y be continuous random variables with joint probability density function f(x,y) distributions for X and Y are (i/3) (x +y), for (x, y) in the rectangular region 0ss1,0Sys 2. The two marginal Ix(x)- (z+1), if 0 251 fy(y) = (1+2y), if0 y 2 Calculate E(x IY -v) and Var (X |Y ) for each y l0,2).
Set Theory and Conditional Probability Problem #1 : (10pts) If P(A) 0.3 and P(B)0.2 and P(A n B) - 0.1. Determine the following probabilities Problem #2: (10pts) a) If the sets Xand Yare non-mutually exclusive , show that: b) Given two events X and Y, draw a Venn diagram to demonstrate that P(X)- P(XnY) + P(XnY), and deduce that P(X)- P(X/Y)P(Y) + P(X/Y)P(Y). Problem #3: (15pts) Consider two events X and Y with probabilities, P(X) 7/15, P(XnY)-1/3, and P(X/Y)-2/3. Calculate...
1. A difference equation is shown below. y(n)- -0.25 y(n-1)+ 0.125 y(n-2)+ x(n)+x(n-1) (a) Find the transfer function H(z) = Y(z)/ X(z) (b) Find Y(z) ifx(n) = (0.4)nu(n) (n=0,1,2,3, ) (c) If x(n) = y(n)-0 for all n < 0, calculate the values of y(0), y(1) and y(2) directly from the difference equation.
A statistical program is recommended. Consider the following data for two variables, x and y. x 22 24 26 30 35 40 y 13 20 33 35 40 36 (a) Develop an estimated regression equation for the data of the form ŷ = b0 + b1x. (Round b0 to one decimal place and b1 to three decimal places.) ŷ = (b) Use the results from part (a) to test for a significant relationship between x and y. Use α =...
Consider an LTI system with input sequence x[n] and output sequence y[n] that satisfy the difference equation 3y[n] – 7y[n – 1] + 2y[n – 2] = 3x[n] – 3x[n – 1] (2.1) The fact that sequences x[ ] and y[ ] are in input-output relation and satisfy (2.1) does not yet determine which LTI system. a) We assume each possible input sequence to this system has its Z-transform and that the impulse response of this system also has its Z-transform. Express the...
Problem 3. Let X and Y be two independent random variables taking nonnegative integer values (a) Prove that for any nonnegative integer m 7m k=0 b) Suppose that X~ B (n, p) and Y ~ B(m. p), and X, Y are independent. What is the distribution of the random variable Z X + Y? (c) Prove the following formula for binomial coefficients: n\ _n + m for kmin (m, n) (d) Let X ~ B (n, 1/2). What is P...
Consider the difference equation y(n) = -0.6y(n-1) - x(n-1) for n>=0 and y(-1) = 0 1. WITHOUT using Laplace or Z-Transform, determine the unit-impulse response h(n) starting from h(0).
Question 3. Separation of variables Consider Laplace's Equation in two dimensions (a) Write Ф(r,y)-F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region {(x, y) E R2: 0Ka, 0 y b with three boundary conditions on Ф об obtain conditions on F and G on those boundaries where conditions on Ф are given (c) (i) Solve the differential equations found in (a), subject to the conditions found in (b)...
1. Let X and Y be two independent random variables following beta distributions Beta(120, 2019) (a) What's P(X 0.3)? (b) What's E(2X - Y)? (c) What's P(2X +4 > 3Y)? (d) What's P(X < Y)? (e) Now if X and Y are no longer independent to each other. Will the answers to a)-(d) remain the same? Explain. (f) Now define Z~Beta(2019, 120). Compare the median of X and Z, which one is bigger? Compare the variance of X and Z,...
Question 3. Separation of variables. Consider Laplace's Equation in two dimensions: (a) Write Φ(x,y) F(x)G(y) and use separation of variables to get ordinary differential equa- tions for F and G (b) Consider the rectangular region {(x,y) є R2 : 0 a, 0-y-b} with three boundary x conditions on Ф: obtain conditions on F and G on those boundaries where conditions on Ф are given. (c) (i) Solve the differential equations found in (a), subject to the conditions found in (b)...