The waiting time between events has **exponential** distribution with parameter $\lambda$. Let $X_1$ be the waiting time until the first event, $X_2$ be the waiting time between the first event and the second, and so on.
Let $Y=X_1+X_2+\cdots+X_{10}$. We want $E(Y)$. By the linearity of expectation, we have $$E(Y)=E(X_1+\cdots+X_{10})=E(X_1)+\cdots +E(X_{10})=\frac{10}{\lambda}.$$