The missing regressor

Idea. The left-out factor $X_2$ is correlated with $X_1$, so part of its effect is credited to $X_1$. The size of that extra part is the true coefficient of $X_2$ times the slope of $X_2$ on $X_1$.

Step 1: covariance. By bilinearity and independence of the noise, $$\begin{array}{rl}\mathrm{Cov}(X_1,Y)& =2\mathrm{Var}(X_1)+3\mathrm{Cov}(X_1,X_2)+0\\ & =2+3\cdot 0.5=\mathrm{3.5.}\end{array}$$

Step 2: slope. $\mathrm{Var}(X_1)=1$, so the slope converges to $3.5=\frac{7}{2}$.

The general formula. If $Y={\beta }_1{X}_1+{\beta }_2{X}_2+\epsilon$ and $X_2$ is omitted, the slope on $X_1$ converges to $${\beta }_1+{\beta }_2\delta ,\delta =\frac{\mathrm{Cov}(X_1,X_2)}{\mathrm{Var}(X_1)}.$$ Here the bias is $3\cdot 0.5=1.5$. The bias vanishes only if the omitted variable has no effect (${\beta }_2=0$) or is uncorrelated with the included one ($\delta =0$). A large sample does not help: the estimate is consistent for the wrong number.