1.4 Exercise 21 — Applied Probability

Suppose $P_{1} , P_{2}$ are probability measures and $c_{1} , c_{2}$ are positive numbers such that $c_{1} + c_{2} = 1$ . Show that the assignment $P \left(E\right) = c_{1} P_{1} \left(E\right) + c_{2} P_{2} \left(E\right)$ to the class of events is a probability measure. Such a combination of probability measures is known as a mixture. Extend this to

P \left(E\right) = \sum_{i = 1}^{n} c_{i} P_{i} \left(E\right) , \text{where} \text{the} P_{i} \text{are} \text{probabilities} \text{measures}, c_{i} > 0 , \text{and} \sum_{i = 1}^{n} c_{i} = 1

Pfeiffer, Applied Probability, 1.4 Problems on Probability Systems, exercise 21

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