Continuous Random Variables, Probability Density Functions, and Statistical Measures in Business Calculus

Continuous Random Variables

Definition and Properties

Continuous random variables are fundamental in probability theory and statistics, especially in business calculus where integration is used to compute probabilities and expected values. Unlike discrete random variables, which take on countable values, continuous random variables can assume any value within a given interval.

• Continuous Random Variable: A variable whose possible values form an interval or a collection of intervals on the real number line.

• Probability is assigned to intervals, not individual points.

• Examples: Height, time, shelf life of a product.

Probability Density Function (PDF)

Definition and Properties

The probability density function (PDF) describes the likelihood of a continuous random variable taking on a particular value. The PDF is a non-negative function whose integral over the entire space equals 1.

• Definition: For a continuous random variable $X$, the PDF $f(x)$ satisfies:

  • $f(x) \geq 0$ for all $x$

  • $\int_{-\infty}^{\infty} f(x) dx = 1$

  • The probability that $X$ lies in $[a, b]$ is $P(a \leq X \leq b) = \int_a^b f(x) dx$

• Probability at a single point: $P(X = a) = 0$

Example: For $f(x) = kx^2$ on $[0,1]$, find $k$ so that $f(x)$ is a PDF:

• $\int_0^1 kx^2 dx = 1 \implies k \int_0^1 x^2 dx = 1 \implies k \cdot \frac{1}{3} = 1 \implies k = 3$

Cumulative Distribution Function (CDF)

Definition and Properties

The cumulative distribution function (CDF) $F(x)$ gives the probability that the random variable $X$ is less than or equal to $x$. It is obtained by integrating the PDF from $-\infty$ to $x$.

• Definition: $F(x) = P(X \leq x) = \int_{-\infty}^x f(t) dt$

• Properties:

  • $F(-\infty) = 0$, $F(\infty) = 1$

  • $F(x)$ is non-decreasing

  • $F(x)$ is continuous for continuous random variables

Example: For $f(x) = \frac{1}{2}$ for $1 \leq x \leq 3$, $F(x)$ is:

• If $x < 1$, $F(x) = 0$

• If $1 \leq x \leq 3$, $F(x) = \int_1^x \frac{1}{2} dt = \frac{x-1}{2}$

• If $x > 3$, $F(x) = 1$

Applications of PDF and CDF

Solving Probability Problems

Business calculus often involves finding probabilities for intervals, expected shelf life, or other measures using the PDF and CDF.

• Example: The shelf life $X$ (in days) of a drug has PDF $f(x) = \frac{200}{1000 + x^2}$ for $x \geq 0$.

• To find $P(X \leq 20)$, compute $\int_0^{20} f(x) dx$.

• To find $P(X > 110)$, compute $\int_{110}^{\infty} f(x) dx$.

• To find $x$ such that $P(X \leq x) = 0.8$, solve $\int_0^x f(t) dt = 0.8$.

Expected Value, Standard Deviation, and Median

Statistical Measures for Continuous Random Variables

Statistical measures such as expected value (mean), variance, standard deviation, and median are essential for summarizing the behavior of random variables in business contexts.

• Expected Value (Mean): $\mu = E[X] = \int_{-\infty}^{\infty} x f(x) dx$

• Variance: $V(X) = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx$

• Standard Deviation: $\sigma = \sqrt{V(X)}$

• Alternative Formula for Variance: $V(X) = \int_{-\infty}^{\infty} x^2 f(x) dx - \mu^2$

• Median: The value $m$ such that $P(X \leq m) = 0.5$

Example: For $f(x) = \frac{1}{2}$ for $1 \leq x \leq 3$:

• $\mu = \int_1^3 x \cdot \frac{1}{2} dx = 2$

• $V(X) = \int_1^3 x^2 \cdot \frac{1}{2} dx - \mu^2 = \frac{8}{3} - 4 = \frac{8}{3} - 4$

• $\sigma = \sqrt{V(X)}$

Worked Examples

Example: Shelf Life of a Drug

• PDF: $f(x) = \frac{200}{1000 + x^2}$ for $x \geq 0$

• Find: Probability that shelf life exceeds 110 days: $P(X > 110) = \int_{110}^{\infty} f(x) dx$

• Find: Value $x$ such that $P(X \leq x) = 0.8$

Example: Daily Electricity Consumption

• PDF: $f(x) = 0.2 - 0.02x$ for $0 \leq x \leq 10$

• Find: Expected daily consumption: $E[X] = \int_0^{10} x f(x) dx$

• Find: Median daily consumption: Solve $P(X \leq m) = 0.5$

Summary Table: Key Properties of PDF and CDF

Property	PDF ($f(x)$)	CDF ($F(x)$)
Definition	Describes likelihood at each value	Probability $X \leq x$
Range	$f(x) \geq 0$	$0 \leq F(x) \leq 1$
Integral	$\int_{-\infty}^{\infty} f(x) dx = 1$	$F(x) = \int_{-\infty}^x f(t) dt$
Probability for interval $[a, b]$	$\int_a^b f(x) dx$	$F(b) - F(a)$

Conclusion

Continuous random variables, probability density functions, and cumulative distribution functions are essential concepts in business calculus, especially for modeling and analyzing real-world phenomena such as product shelf life and resource consumption. Mastery of these topics enables students to compute probabilities, expected values, variances, and medians using integration techniques.