# Expected value of chi-square distribution

#### Jorris

##### New Member
The chi-square distribution is defined as the distribution of
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?

#### katxt

##### Active Member
X_1 and X_1 are not independent so E(X_1**2) does not = E(X_1)*E(X_1)