The time (in minutes) required to complete an assembly on a...

The time (in minutes) required to complete an assembly on a production line is a random variable $X$ with the cumulative distribution function $F(x)=\frac{1}{123} x^{3}$ $0 \leq x \leq 5.$ (a) Find $E(X)$ and give an interpretation of this quantity. (b) Compute $\operatorname{Var}(X).$

The time (in minutes) required to complete an assembly on a production line is a random variable $X$ with the cumulative distribution function $F(x)=\frac{1}{123} x^{3}$ $0 \leq x \leq 5.$
(a) Find $E(X)$ and give an interpretation of this quantity.
(b) Compute $\operatorname{Var}(X).$

Key Concepts

Variance

Variance is a measure of the dispersion or spread of the random variable around its mean. It quantifies how much the outcomes differ from the expected value by averaging the squared differences between the outcomes and the mean. A larger variance indicates greater spread, while a smaller variance suggests outcomes are more tightly clustered around the mean. This is important for assessing the reliability and risk associated with the random variable’s outcomes.

Cumulative Distribution Function (CDF)

The cumulative distribution function is a fundamental concept in probability theory that defines the probability that a random variable will take a value less than or equal to a given number. It is a non-decreasing function that provides a complete description of the distribution of a random variable, and is used as the starting point for deriving other properties of the distribution, such as the probability density function and moments.

Probability Density Function (PDF)

For continuous random variables, the probability density function is obtained by differentiating the cumulative distribution function. The PDF represents the relative likelihood of the random variable taking on a specific value. Integrals of the PDF over intervals yield probabilities, and it plays a crucial role in computing expected values, variances, and other moments of the distribution.

Expected Value

The expected value, or mean, of a random variable is a measure of its central tendency, representing the long-run average outcome if the random process were repeated many times. It is calculated by taking a weighted average of all possible values, where the weights are determined by the probability density. In practical terms, the expected value provides a single summary statistic that reflects the typical outcome of the random variable.

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