Question

The probability distribution of \(x,\) the number of defective tires on a randomly selected automobile checked at a certain inspection station, is given in the following table: $$ \begin{array}{lccccc} x & 0 & 1 & 2 & 3 & 4 \\ p(x) & 0.54 & 0.16 & 0.06 & 0.04 & 0.20 \end{array} $$ a. Calculate the mean value of \(x\). b. Interpret the mean value of \(x\) in the context of a long sequence of observations of number of defective tires. c. What is the probability that \(x\) exceeds its mean value? d. Calculate the standard deviation of \(x\).

Short answer

The mean value is given by the formula \(\mu = \sum x\cdot p(x)\) which represents the expected number of defective tires. Probability that \(x\) exceeds its mean is the sum of probabilities for the outcomes greater than the mean. Standard deviation is given by \(\sigma = \sqrt{\sum (x-\mu)^2 \cdot p(x)}\).

Step-by-step solution

Calculate Mean Value

The mean of a probability distribution can be found by summing the product of each outcome and their respective probabilities. Expression for mean in this context is \(\mu = \sum x\cdot p(x)\). So, \(\mu = 0*0.54 + 1*0.16 + 2*0.06 + 3*0.04 + 4*0.20.\)

Interpret the Mean Value

The mean value, \(\mu\), represents the expected value or 'average' number of defective tires per automobile. This is the average number of defective tires that would be seen if a large number of automobiles were inspected.

Calculate the Probability that \(x\) exceeds its mean value

We need to identify which values of \(x\) exceed the mean and then sum their corresponding probabilities, \(p(x)\). This will give the probability of \(x\) exceeding its mean value.

Calculate Standard Deviation

Standard deviation can be computed with the formula \(\sigma = \sqrt{\sum (x-\mu)^2 \cdot p(x)}\), where \(x\) is each possible outcome, \(p(x)\) is the respective probability of that outcome and \(\mu\) is the mean value calculated earlier.

Key concepts

Mean Value of a Distribution

In probability and statistics, the mean value of a distribution is a measure of the central tendency of the distribution. It is calculated by taking the sum of the product of each possible outcome and their respective probabilities. For a discrete probability distribution, the formula for the mean value, often denoted as \(\text{\(\mu\)}\), is

\[\mu = \sum x \cdot p(x)\]
where \(x\) represents the value of each possible outcome and \(p(x)\) represents the probability of that outcome. In our example of defective tires, the mean value is calculated as \(\text{\(0\times0.54 + 1\times0.16 + 2\times0.06 + 3\times0.04 + 4\times0.20\)}\). By performing this calculation, we can find the average or 'expected' outcome of the distribution.

Interpretation of Mean Value

The interpretation of the mean value in a real-world context provides insight into what we can expect on average over a large number of trials. For our automobile inspection example, the mean value represents the average number of defective tires per car. If you were to inspect a very large number of cars at the station, the mean value is the number of defective tires you would expect to find on an average car. It's important not to interpret the mean value as the most common outcome; rather, it is the arithmetic average of all possible outcomes, weighted by their probabilities.

Probability Calculations

Performing probability calculations involves determining the likelihood of various outcomes. To calculate the probability that the random variable \(x\) exceeds its mean value, we look at the probabilities of the outcomes greater than the mean. For instance, if the calculated mean value is 1.2, we consider the probabilities for all values of \(x\) that are greater than 1.2, such as 2, 3, and 4. Then, we add these probabilities together to find the overall likelihood that \(x\) will exceed the mean. This process requires a clear understanding of the probability distribution and its outcomes.

Standard Deviation

The standard deviation is a measure of the amount of variability or spread in a set of values. It is denoted by \(\text{\(\sigma\)}\) and for our distribution, it is calculated using the formula

\[\sigma = \sqrt{\sum (x-\mu)^2 \cdot p(x)}\]
where \(x\) is the outcome value, \(\mu\) is the mean value, and \(p(x)\) is the probability of the outcome. To compute the standard deviation for the number of defective tires, we substitute the mean value we found earlier into the formula and perform the calculation. This will give us an indication of how much the number of defective tires diverges from the mean on average, informing us about the distribution's consistency.