A salesman has scheduled two appointments to sell...

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability $.3,$ and his second will lead independently to a sale with probability .6. Any sale made is equally likely to be either for the deluxe model, which costs $\$ 1000$, or the standard model, which costs $\$ 500 .$ Determine the probability mass function of $X,$ the total dollar value of all sales.

Key Concepts

Probability Mass Function

A probability mass function (PMF) describes the probability that a discrete random variable is exactly equal to some value. It assigns a probability to each potential outcome of the variable such that the sum of the probabilities is one. Understanding the PMF is key for analyzing discrete outcomes and making further probabilistic inferences based on the assigned probabilities.

Independent Events

Independent events are events whose occurrence or non-occurrence does not affect the probability of occurrence of the other event. In many probability problems, assuming independence allows for the calculation of joint probabilities by multiplying the probabilities of the individual events.

Discrete Random Variables

A discrete random variable is one that can take on a countable number of distinct values. When modeling real-world situations, such as the outcomes of certain events or transactions, discrete random variables are used to represent and analyze probabilities for those distinct outcomes.

Joint Distribution

The joint distribution refers to the probability distribution of two or more random variables considered simultaneously. When random variables are independent, their joint probability can typically be calculated as the product of their individual probabilities, allowing for the analysis of combined outcomes.

Transformation of Random Variables

The transformation of random variables involves deriving a new random variable from one or more existing ones through some function. In cases where outcomes are combined, such as summing values from independent sales, the resulting distribution is obtained by considering all possible combinations of outcomes and their probabilities.

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