$\{X(t), t ≥ 0 \}$ is a pure birth process with $X(0) = 0$.
Its birth rates $λk$ are such that $λk=8−k$ for $0≤k<8$ and $λk=1$ for $k≥8$.
Let $Wk =min\{t:X(t)=k\}$ be the waiting times.
Find the variance of $W_1+W_2+⋯+W_8$.
My approach is the following: The problem is asking for $Var(W1+W2+⋯+W8)$, this could be written as $Var(W_1)+ Var(W_2) + ⋯ + Var(W_8)$.
The $Wk$ distributions all follow an exponential distribution with parameter $λk$, so each $E(X_o) = \frac{1}{λ_0}$ and $Var(X_o) = \frac{1}{λ_0^2}$ etc.
$$Var(W_1) = Var(X_0) = \frac{1}{(λ_0)^2}$$
$$Var(W_2) = Var(X_0 + X_1) = \frac{1}{(λ_0)^2} + \frac{1}{(λ_1)^2}$$
$$Var(W_3) = Var(X_0 + X_1 + X_2) = \frac{1}{(λ_0)^2} + \frac{1}{(λ_1)^2} + \frac{1}{(λ_3)^2}$$
So on, so for
$$Var(W_1+W_2+⋯+W_8) = Var(W_1)+ Var(W_2) + ⋯ + Var(W_8) $$ $$= Var(X_0) + (Var(X_0 + X_1)) + ... Var(X_0 + X_1 + X_2 ... X_8) $$ $$= 8 \times Var(X_0) + 7 \times Var(X_1) + Var(X_8) $$
$$= 8 \times \frac{1}{(λ_0)^2} + 7 \times \frac{1}{(λ_1)^2} ... + \frac{1}{(λ_8)^2}$$,
so I have $2.717$
However, the multiple choice only has integers. What am I doing wrong?