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I have 2 random variables, all independant, discrete and uniform

I want to compute standard deviation for Z = X - Y

Here is the R code:

#create X and Y
X = rep(0:1, c(733-41,41))
Y = rep(0:1, c(742-60,60))

#create Z as Y - X
Z = c()
for (i in X) {Z = c(Z,c(Y-i))}

Means are equal:

mean(Z)
[1] 0.02492802
mean(Y) - mean(X)
[1] 0.02492802

Then I count standard deviation and variance of Z using the formula:

sd(Y) + sd(X)
[1] 0.5027599
var(Y) + var(X)
[1] 0.1273021

Ann then, using sd() and var() functions:

sd(Z)
[1] 0.3565528
var(Z)
[1] 0.1271299

Where is my mistake and why sd(Y) + sd(X) != sd(Z) and var(Z) != var(Y) + var(X)?

Thanks!

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