Expectations of outcomes

Question

$X_1, X_2, \ldots,X_{50}$ represents the 50 outcomes of 50 independent fair coin flips. $$P(X_i=0)=P(X_i=1)=0.5$$ I am trying to analyze the different possibilities of $\operatorname{E}(X)$ and $\operatorname{var}(X)$. Define $X$ to be $X = X_1 - X_2.$ For instance, why would the answers for $\operatorname{E}(X_1)$ and $\operatorname{E}(X)$ be different? Similarly for $\operatorname{var}(X_1)$ and $\operatorname{var}(X)$. My confusion stems how to even interpret $\operatorname{E}(X_1)$. Does it mean what is the probability of getting the same outcome as $X_1$? and if so, how would I interpret $\operatorname{E}(X)$?

Answer 1

Hint:

Expectation is linear, i.e. \begin{align} E[aX+bY] = aE[X]+bE[Y]. \end{align}

Since $X_i$ are independent then it follows \begin{align} \operatorname{Var}[aX_1+bX_2] = a^2\operatorname{Var}X_1+b^2\operatorname{Var}X_{2}. \end{align}