Question

Q2: Suppose that X-N(O, 1), U-N(O, 0.25), Y 3- 2X and Z following questions. 2 X +U. Please answer the Compute E(Y), E(Z), Var(Y) and Var(Z). What are the distributions of Y and Z? Using R, draw 50 independent realizations of X and U. Using those values, create 50 realizations of Y and Z. (NOTE: set the seed for random number generation in R. Before your code type set.seed 123))

Answer 1

E[Y] = E[3-2X] = 3-2E[X] = 3 - 2 * 0 = 3

E[Z] = E[2 - X + U] = 2 - E[X] + E[U] = 2 - 0 + 0 = 2

Var[Y] = Var[3-2X] = 0 + (-2)² Var[X] = 4 * 1 = 4

Var[Z] = Var[2 - X + U] = 0 +  (-1)² Var[X] + 1²Var[U] = 1 + 0.25 = 1.25

We know that the linear combination of normal random variables is a random variable. Thus, the distributions of Y and Z are

Y ~ N(3, 4) and Z ~ N(2, 1.25)

Run the below code in R to draw 50 independent realizations of X, U, Y and Z

set.seed(123)
x = rnorm(50, mean = 0, sd = sqrt(1))
u = rnorm(50, mean = 0, sd = sqrt(0.25))
y = 3 - 2*x
z = 2 - x + u

I get the below outputs,

> x
[1] -0.56047565 -0.23017749 1.55870831 0.07050839 0.12928774 1.71506499 0.46091621 -1.26506123
[9] -0.68685285 -0.44566197 1.22408180 0.35981383 0.40077145 0.11068272 -0.55584113 1.78691314
[17] 0.49785048 -1.96661716 0.70135590 -0.47279141 -1.06782371 -0.21797491 -1.02600445 -0.72889123
[25] -0.62503927 -1.68669331 0.83778704 0.15337312 -1.13813694 1.25381492 0.42646422 -0.29507148
[33] 0.89512566 0.87813349 0.82158108 0.68864025 0.55391765 -0.06191171 -0.30596266 -0.38047100
[41] -0.69470698 -0.20791728 -1.26539635 2.16895597 1.20796200 -1.12310858 -0.40288484 -0.46665535
[49] 0.77996512 -0.08336907

> u
[1] 0.126659257 -0.014273378 -0.021435229 0.684301142 -0.112885493 0.758235302 -0.774376402 0.292306875
[9] 0.061927122 0.107970784 0.189819741 -0.251161727 -0.166603692 -0.509287692 -0.535895613 0.151764321
[17] 0.224104889 0.026502113 0.461133734 1.025042343 -0.245515583 -1.154584438 0.502869262 -0.354600381
[25] -0.344004308 0.512785685 -0.142386504 -0.610358856 0.090651740 -0.069445681 0.002882093 0.192640201
[33] -0.185330016 0.322188274 -0.110243281 0.165890982 0.548419507 0.217590745 -0.162965793 0.574403809
[41] 0.496751928 0.274198480 0.119365868 -0.313953038 0.680326224 -0.300129794 1.093666497 0.766305313
[49] -0.117850180 -0.513210450

> y
[1] 4.1209513 3.4603550 -0.1174166 2.8589832 2.7414245 -0.4301300 2.0781676 5.5301225 4.3737057
[10] 3.8913239 0.5518364 2.2803723 2.1984571 2.7786346 4.1116823 -0.5738263 2.0042990 6.9332343
[19] 1.5972882 3.9455828 5.1356474 3.4359498 5.0520089 4.4577825 4.2500785 6.3733866 1.3244259
[28] 2.6932538 5.2762739 0.4923702 2.1470716 3.5901430 1.2097487 1.2437330 1.3568378 1.6227195
[37] 1.8921647 3.1238234 3.6119253 3.7609420 4.3894140 3.4158346 5.5307927 -1.3379119 0.5840760
[46] 5.2462172 3.8057697 3.9333107 1.4400698 3.1667381

> z
[1] 2.6871349 2.2159041 0.4198565 2.6137928 1.7578268 1.0431703 0.7647074 3.5573681 2.7487800
[10] 2.5536328 0.9657379 1.3890244 1.4326249 1.3800296 2.0199455 0.3648512 1.7262544 3.9931193
[19] 1.7597778 3.4978338 2.8223081 1.0633905 3.5288737 2.3742908 2.2810350 4.1994790 1.0198265
[28] 1.2362680 3.2287887 0.6767394 1.5764179 2.4877117 0.9195443 1.4440548 1.0681756 1.4772507
[37] 1.9945019 2.2795025 2.1429969 2.9548748 3.1914589 2.4821158 3.3847622 -0.4829090 1.4723642
[46] 2.8229788 3.4965513 3.2329607 1.1021847 1.5701586