Q.For the same distribution of $X$ (number of heads in two coin tosses), find the variance $Var(X)$ and the standard deviation $SD(X)$.

Since $E(X)=1$ from the previous example, variance can be found using $Var(X)=E(X^2)-[E(X)]^2$, where $E(X^2)$ is found by weighting the squares of each value by its probability. …

From the earlier examples, the distribution is:

and $E(X) = 1$.

Step 1 — find $E(X^2)$:

Step 2 — apply the shortcut formula:

Step 3 — standard deviation:

Cross-check using the definition directly, $Var(X) = \sum (x_i - \mu)^2 p_i$ with $\mu = E(X) = 1$: …

The direct definitional method $Var(X) = \sum(x_i-\mu)^2 p_i$ shown above as the cross-check can also be used as the primary method; it gives the identical answer of $0.5$ but …

A common mistake is to forget to square $E(X)$ in the shortcut formula and instead subtract $E(X)$ itself — always square the mean before subtracting it from $E(X^2)$. Another mistake is leaving the answer as variance when standard dev …