Traffic Light Wait Time: Using Probability to Decode Delays

Step 1 of 2

(a) Using the total probability theorem, we have

\(\mathbf{P}(X>4)=\sum_{k=0}^{4} \mathbf{P}(k \text { lights are red }) \mathbf{P}(X>4 \mid k \text { lights are red })\).

We have

\(\mathbf{P}(k \text { lights are red })=\left(\begin{array}{l} 4 \\ k \end{array}\right)\left(\frac{1}{2}\right)^{4} \)

The conditional PDF of \(X\) given that \(k\) lights are red, is normal with mean \(k\) minutes and standard deviation \((1 / 2) \sqrt{k}\). Thus, \(X\) is a mixture of normal random variables and the transform associated with its (unconditional) PDF is the corresponding mixture of the transforms associated with the (conditional) normal PDFs. However, \(X\) is not normal, because a mixture of normal PDFs need not be normal. The probability \(\mathbf{P}(X>4 \mid k\) lights are red) can be computed from the normal tables for each \(k\), and \(\mathbf{P}(X>4)\) is obtained by substituting the results in the total probability formula above.