Y = X_1**2 + X_2**2 + ... + X_k**2

with X_i independent standard normally distributed (mean 0, std 1).

The expected value of a sum of variables is

E(X_1 + X_2) = E(X_1) + E(X_2)

The expected value of a product of independent variables is

E(X_1 * X_2) = E(X_1) * E(X_2)

Combining all of this

E(Y) = E(X_1**2 + X_2**2 + ... + X_k**2)

= E(X_1**2) + E(X_2**2) + ... + E(X_k**2)

= E(X_1)*E(X_1) + E(X_2)*E(X_2) + ... + E(X_k)*E(X_k)

= 0*0 + 0*0 + ... + 0*0

= 0

But this is not what I read here: https://www.statlect.com/probability-distributions/chi-square-distribution

E(Y) = k

What did I do wrong?