[NW] [DATA] TelevisionsIn the Sullivan Statistics Survey I, individuals were asked to disclose the number of televisions in their household. In the following probability distribution, the random variable X represents the number of televisions in households.

c. Calculate and explain the mean of the random variable X.

Verified step by step guidance

1. Identify the random variable X, which represents the number of televisions in a household, and the corresponding probabilities P(x) for each value of X from the table.
2. Recall that the mean (or expected value) of a discrete random variable X is calculated using the formula: \[\text{Mean} = E(X) = \sum (x \times P(x))\] where the sum is taken over all possible values of x.
3. Multiply each value of x (number of televisions) by its corresponding probability P(x) from the table. For example, calculate 0 \times 0, 1 \times 0.161, 2 \times 0.261, and so on for all values of x.
4. Add all the products obtained in the previous step to get the sum \( \sum (x \times P(x)) \). This sum represents the mean number of televisions per household.
5. Interpret the mean as the average number of televisions in a household based on the given probability distribution, which provides a measure of central tendency for the random variable X.

Key Concepts

Random Variable

A random variable is a numerical outcome of a random phenomenon. In this context, X represents the number of televisions in a household, which can take on discrete values from 0 to 9. Understanding the random variable helps in analyzing the distribution of possible outcomes.

Probability Distribution

A probability distribution assigns probabilities to each possible value of a random variable, showing how likely each outcome is. Here, P(x) gives the probability that a household has x televisions. The sum of all probabilities must equal 1, ensuring a complete description of all outcomes.

Mean (Expected Value) of a Discrete Random Variable

The mean or expected value is the weighted average of all possible values of the random variable, calculated by summing the products of each value and its probability. It represents the long-run average number of televisions per household in this survey.