Question

A local television station sells 15 -second, 30 -second, and 60 -second advertising spots. Let \(x\) denote the length of a randomly selected commercial appearing on this station, and suppose that the probability distribution of \(x\) is given by the following table: $$ \begin{array}{lrrr} x & 15 & 30 & 60 \\ p(x) & 0.1 & 0.3 & 0.6 \end{array} $$ What is the mean length for commercials appearing on this station?

Short answer

The mean length for commercials appearing on the station is 46.5 seconds.

Step-by-step solution

Identify the Variables and their Probabilities

From the given problem, we have three possible lengths for the commercial, 15, 30, and 60 seconds and the associated probabilities are 0.1, 0.3, and 0.6 respectively. These can be represented as pairs: (15, 0.1), (30, 0.3), (60, 0.6).

Apply the Expected Value Formula

We calculate the mean (expected value) using the formula \(\mu = \Sigma [x * P(x)]\). Plug the values from Step 1 into the formula: \(\mu = (15 * 0.1) + (30 * 0.3) + (60 * 0.6)\)

Calculate the Mean

Carry out the multiplication and addition operations to find the mean: \( \mu = 1.5 + 9 + 36 = 46.5\)

Key concepts

Probability Distribution

Understanding the probability distribution is critical to analyzing data and making predictions about future events. It is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. A probability distribution is like a map that guides us in the landscape of probability, showing where we're likely to find the values of a random variable and with what frequency.

In our exercise, the random variable represents the length of a commercial, and the probability distribution is given by the table, linking each commercial length to its probability. For example, we see that a 60-second commercial is most likely to occur, as it has the highest probability (0.6). When we sum all the probabilities, they should equal 1, confirming that our distribution covers all possible outcomes of the random variable.

Mean

The mean, often known as the expected value in the context of probability, is the long-run average value of repetitions of the experiment it represents. Simply put, it's what you would anticipate as the outcome over an extended period or over many trials. It takes into account all possible values and their probabilities, making it a fundamental measure of the central tendency of a probability distribution.

In calculating the mean length of commercials in our example, we multiplied each length of the commercial by its respective probability and added the values together. This calculation provided us with the average length of a commercial airing on the television station, which is 46.5 seconds.

Random Variable

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types of random variables: discrete and continuous. Discrete random variables, like the one in our textbook example, have a countable number of possibilities (e.g., the lengths of commercials). Continuous random variables, on the other hand, can take an infinite number of values within a range.

In the context of our exercise, the random variable 'x' represents the possible lengths of commercials (15, 30, and 60 seconds), which can be observed with certain probabilities. Random variables are the foundations on which probability distributions are built, and understanding them is essential for working with any statistical model.