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MULUNGUSHI UNIVERSITY
SNAS-Department of Mathematics and Statistics
EGM411 (NUMERICAL COMPUTATION AND STATISTICS)
Tutorial sheet 2
——————————————————————————————————- Question 1. Apply the eigen analysis method, which requires eigenvalues and eigenvectors, to solve the differential system below.
dx dt
= x − z dy dt
= x + 2y + 2z dz dt
= 2x + 2y + 3z
Show all eigen analysis steps and display the differential equation answer in vector form
Question 2.
(a) Consider the system of differential equations given by x′^ = P(t)x with the initial condition x (t 0 ) = x 0. If P(t) =
− 2 t−^2 2 t−^1
, is this a linear or nonlinear problem? If we apply the initial condition, will there be a unique solution? Explain.
(b) If P(t) = B, a 2 × 2 constant real-valued matrix, and a solution to the system
is x =
cos(3t) cos(3t) − 2 sin(3t)
e−^4 t, what are the eigenvalues of B?
(c) Consider the system
x y z
′
=
x y z
. The coefficient ma-
trix has eigenvalues and eigenvectors λ = −2 and λ = ±i, with v− 2 =
and v±i =
±i
. Write a general, real-valued solution to the system.
Hint: Use Euler’s formula: eiθ^ = cos θ + i sin θ.
Question 3. Find explicit, real-valued solutions to
x′^ =
x, x(0) =
Hint: Simplify further by using Euler’s formula: eiθ^ = cos θ + i sin θ.
Question 4. Find the Z-transform of the following signals a) x[n] = u[n] b) x[n] = 2(0.8)nu[n] c) x[n] = [2 (3n) − 4 (3n)] u[n] d) x[n] = [0. 5 n^ + (− 0 .4)n] u(n) e) x[n] = − 0. 5 nu(n − 2) + (− 0 .4)nu(n) f) x[n] = 0. 8 n[u(n) − u(n − 10)] g) x[n] = 7
3
n u[n] − 6
2
n u[n].
Question 5. Let
X(z) =
z + 3 (z − 0 .7)(z + 0.5)
, |z| > 0. 7.
a) Use partial fraction expansion to determine x[n]. b) If the region of convergence is instead given by |z| < 0 .5, what would your answer for part (a) be?
Question 6. Find the inverse Z-transforms of the following signals a)
X(z) =
(z − 1)(z + 0.8) (z − 0 .5)(z + 0.2)
b)
X(z) =
(z + 0.8) (z − 0 .5)(z + 0.2)
c)
X(z) =
z^3 + z + 1 (z^2 − 0. 5 z + 0.25) (z − 1)
d)
X(z) =
z(z + 1) (z − 1) (z^2 − z + 1/4)
Question 7.
Country Cigarette Coronary United States 3900 259. 9 Canada 3350 211. 6 Australia 3220 238. 1 New Zealand 3220 211. 8 United Kingdom 2790 194. 1 Switzerland 2780 124. 5 Ireland 2770 187. 3 Iceland 2290 110. 5 Finland 2160 233. 1 West Germany 1890 150. 3 Netherlands 1810 124. 7 Greece 1800 41. 2 Austria 1770 182. 1 Belgium 1700 118. 1 Mexico 1680 31. 9 Italy 1510 114. 3 Denmark 1500 144. 9 France 1410 144. 9 Sweden 1270 126. 9 Spain 1200 43. 9 Norway 1090 136. 3
a)Calculate the Pearson correlation coefficient and interprate b)Based on the scatterplot of Coronary versus Cigarette, does there appear to be a linear relationship between cigarette consumption and heart disease? If so, does the relationship appear to be negative or positive? c) Write the equation for the fitted model. d) Give an interpretation of the fitted slope, βˆ. e) How much natural variability is associated with the estimated intercept ˆα? f) Compute the residual for Greece. h) Do you think that natural variability alone could account for such a large value of βˆ as actually found here? Explain. g) Determine whether sufficient statistical evidence exists to conclude that there is a positive linear relationship between Cigarette and Coronary at the 1% level of signif- icance.
