A manufacturing company employs two devices to inspect...

A manufacturing company employs two devices to inspect output for quality control purposes. The first device is able to accurately detect $99.3 \%$ of the defective items it receives, whereas the second is able to do so in $99.7 \%$ of the cases. Assume that four defective items are produced and sent out for inspection. Let $X$ and $Y$ denote the number of items that will be identified as defective by inspecting devices 1 and $2,$ respectively. Assume that the devices are independent. Determine: (a) $f_{X Y}(x, y)$ (b) $f_{X}(x)$ (c) $E(X)$ (d) $f_{Y \mid 2}(y)$ (e) $E(Y \mid X=2)$ (f) $V(Y \mid X=2)$ (g) Are $X$ and $Y$ independent?

A manufacturing company employs two devices to inspect output for quality control purposes. The first device is able to accurately detect $99.3 \%$ of the defective items it receives, whereas the second is able to do so in $99.7 \%$ of the cases. Assume that four defective items are produced and sent out for inspection. Let $X$ and $Y$ denote the number of items that will be identified as defective by inspecting devices 1 and $2,$ respectively. Assume that the devices are independent. Determine:
(a) $f_{X Y}(x, y)$
(b) $f_{X}(x)$
(c) $E(X)$
(d) $f_{Y \mid 2}(y)$
(e) $E(Y \mid X=2)$
(f) $V(Y \mid X=2)$
(g) Are $X$ and $Y$ independent?

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, with a constant probability of success on each trial. In quality control contexts, if each item is inspected independently with a known detection rate (probability of identifying a defective item), then the number of items identified as defective follows a binomial distribution.

Joint Probability Mass Function (Joint PMF)

The joint PMF describes the probability that two or more discrete random variables simultaneously take specific values. When the random variables are independent and each follows its own binomial distribution, the joint PMF is given by the product of their individual (marginal) binomial PMFs.

Marginal Distribution

A marginal distribution is the probability distribution of a subset of random variables, obtained by summing or integrating over the probabilities of the other variables in a joint distribution. In problems involving multiple devices or processes, each device's output is described by its own marginal distribution, which can be derived from the joint PMF if the devices are independent.

Conditional Distribution

The conditional distribution specifies the probability distribution of one random variable given that another variable takes a certain value. It is calculated by dividing the joint probability of the variables by the marginal probability of the given variable. This concept is essential when assessing the behavior or performance of one inspection device contingent on the outcome of another.

Expected Value

The expected value of a random variable is a measure of the central tendency or the long-run average outcome of a random process. For binomial distributions, the expected number of successes is the product of the number of trials and the probability of success per trial, and this concept is widely used in reliability and quality control assessments.

Variance

Variance measures the spread or dispersion of a random variable around its mean, indicating the degree of uncertainty or variability in the outcomes. For a binomially distributed variable, the variance is computed as the product of the number of trials, the probability of success, and the probability of failure, providing important insights into the consistency of quality control measures.

Independence of Random Variables

Independence between random variables means that the occurrence of one does not affect the probability distribution of the other. In the context of quality control devices, if the inspections are performed independently by two different devices, then the number of defective items detected by one device does not influence the number detected by the other. This property simplifies the analysis of the joint and marginal distributions.

Please give Ace some feedback