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In probability theory, a probability distribution provides a mathematical function that describes the likelihood of different outcomes in an experiment. When dealing with a discrete random variable, like the number of defects in a car, each possible outcome, denoted as \(x\), has a corresponding probability \(p(x)\). These probabilities must satisfy the following rules:

• Each probability \(p(x)\) must be between 0 and 1.
• The sum of all probabilities must equal 1.

For the car defect example, the probability distribution is given as a table where each defect number has a probability of occurrence. When checking that this table constitutes a valid probability distribution, sum all the probabilities:\[0.04 + 0.20 + 0.34 + 0.20 + 0.15 + 0.04 + 0.03 = 1\]This confirms it is a valid probability distribution, as the sum equals 1.

The expected value, or mean, of a random variable is a measure of the central tendency of the distribution. It gives us an idea of the average outcome if we were to repeat the experiment multiple times. For a discrete random variable \(X\), the expected value \(E(X)\) is calculated by:\[E(X) = \sum_{x} x \, p(x)\]However, in more complex distributions like those involving moment-generating functions (MGF), the expected value can also be derived by differentiating the MGF and evaluating at \(t = 0\). For our car defects example, after computing the MGF, differentiating gives us:\[M'_{X}(0) = 2.46\]Thus, the expected number of defects, \(E(X)\), is 2.46. This means that, on average, you can expect 2.46 defects per car.

Variance is a measure of how much the values of a random variable \(X\) deviate from the mean \(E(X)\). It provides insight into the spread or dispersion within the distribution. The variance \(V(X)\) of \(X\) can be determined using the formula:\[V(X) = E(X^2) - [E(X)]^2\]Here, \(E(X^2)\) represents the expected value of the square of the random variable. After obtaining the MGF and its derivatives for our defects example, we calculate \(E(X^2)\) as:\[M''_{X}(0) = 7.84\]Then, the variance is computed:\[V(X) = 7.84 - (2.46)^2 = 1.74\]The variance, 1.74, illustrates the extent to which defect counts vary from the mean. A lower variance indicates that defect counts are close to 2.46 defects, while a higher variance would indicate more variability.