Suppose your time of travel is 50 minutes and your train is delayed. You also have waiting times at three traffic lights.
Thus, the total time of travel is $W = 50 + B \cdot V + \sum_{i=1}^{3} B_i \cdot A_i$.
All random variables $B, B_1, B_2, B_3, A_1, A_2, A_3, V$ are independent.
We have $B \sim Binomial(1, \frac{1}{10})$, $B_1, B_2, B_3 \sim Binomial(1, \frac{1}{2})$, $A_1, A_2, A_3 \sim Uniform([0, 5])$, $V \sim Exponential(\frac{1}{10})$.
How does one now calculate the expected value $\mathbb{E}(W)$?
$\mathbb{E}(W) = \mathbb{E}(50+B \cdot V + B_1 \cdot A_1 + B_2 \cdot A_2 + B_3 \cdot A_3)$.
Best Answer
Guide:
Ingredients that you need are
Expectations are linear, that is the expecation of the sum is equal to the sum of the expectation$$\mathbb{E}(\sum_{i=1}^n X_i) = \sum_{i=1}^n\mathbb{E}( X_i).$$
Also, use the property that the random variables are independent. For example, $$\mathbb{E}(BV)=\mathbb{E} (B) \mathbb{E}(V).$$
You should be then be able to compute the expectation of each random variable and evaluate them.