What's wrong with the first 2. If substitution of variable is what we are doing then it should be 1. I'm unable to get why only 3rd one is correct. Please help. https://i.stack.imgur.com/Qjbbk.jpg
2026-04-01 17:31:31.1775064691
Expectation Problem.
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Let $X$ be a discrete random variable with probability mass function $p(x)$. Then the expected value of a function $f(x)$ (if it is defined) is $\mathbb{E}[f(x)]=\int_{-\infty}^{\infty}f(x)p(x)dx$. In your case, $f(x)=x^2$, hence only the third formula in your notes is correct.