Continuous Random Variables and Uniform Distributions

Continuous Random Variables

Discrete vs. Continuous Random Variables

Random variables can be classified as either discrete or continuous. Discrete random variables take on countable values, while continuous random variables can take on any value within a given interval.

• Discrete Random Variable (R.V.): Cannot be broken down further; values are distinct and separate.

• Continuous Random Variable (R.V.): Can take on any value within a range; values are not countable but measurable.

For continuous random variables, probabilities are calculated using the probability density function (PDF), denoted as $f(x)$.

• Probability Density Function (PDF): A function that describes the likelihood of a random variable to take on a particular value. For a continuous random variable $X$, the probability that $X$ falls within an interval $[a, b]$ is given by:

$P(a \leq X \leq b) = \int_{a}^{b} f(x) \, dx$

Comparing Discrete and Continuous Probability

The following table summarizes the differences between discrete and continuous probability distributions:

Discrete Probability	Continuous Probability Density
Probability assigned to specific values Sum of probabilities equals 1 Example: $P(X = 2)$	Probability assigned to intervals Total area under the curve equals 1 Example: $P(2 \leq X \leq 5)$

Uniform Distribution

Definition and Properties

The Uniform Distribution is a type of continuous probability distribution where every value within a specified interval is equally likely. The probability density function is constant across the interval.

• Probability Density Function for Uniform Distribution:

$f(x) = \frac{1}{b - a}$ for $a \leq x \leq b$

• Total Area: The area under the PDF curve over the interval $[a, b]$ is 1.

• Probability Calculation: For $X$ uniformly distributed between $a$ and $b$, the probability that $X$ falls between $c$ and $d$ ($a \leq c < d \leq b$) is:

$P(c \leq X \leq d) = \int_{c}^{d} f(x) \, dx = \frac{d - c}{b - a}$

• Example: If $X$ is uniformly distributed between 2 and 12, then $f(x) = \frac{1}{12 - 2} = 0.1$ for $2 \leq x \leq 12$.

Applications and Examples

• Call Center Response Time: If the time to answer a call is uniformly distributed between 2 and 12 seconds, the probability that a call is answered in 5-9 seconds is:

$P(5 \leq X \leq 9) = \frac{9 - 5}{12 - 2} = \frac{4}{10} = 0.4$

• Passenger Arrival Example: If a passenger arrives at a station every 20 minutes, and arrival time is uniformly distributed, the probability that the passenger will wait less than 10 minutes is:

$P(X < 10) = \frac{10}{20} = 0.5$

Probability Density Function (PDF) Validity

Criteria for a Valid PDF

To be a valid probability density function, a curve must satisfy:

• 1. $f(x) \geq 0$ for all $x$

• 2. The total area under the curve equals 1: $\int_{-\infty}^{\infty} f(x) \, dx = 1$

Practice: Given a graph, check if the area under the curve is 1 to determine if it is a valid PDF.

Graphical Representation of Uniform Distribution

Interpreting Probability Density Graphs

For a uniform distribution, the PDF is a horizontal line over the interval $[a, b]$. The height of the line is $\frac{1}{b - a}$, and the area under the curve for any subinterval represents the probability for that interval.

• Example: For $X$ uniformly distributed between 2 and 12, the PDF is a horizontal line at $0.1$ from $x = 2$ to $x = 12$.

• Shading Areas: To find the probability for an interval, shade the region under the curve between the desired bounds and calculate the area.

Summary Table: Discrete vs. Continuous Probability

Aspect	Discrete Probability	Continuous Probability Density
Values	Countable, distinct	Any value in interval
Probability Assignment	Specific values	Intervals (area under curve)
Sum/Area	Sum equals 1	Total area equals 1
Function	Probability mass function (PMF)	Probability density function (PDF)

Additional info: The notes provide foundational concepts for continuous random variables and uniform distributions, which are essential for understanding probability density functions and their applications in statistics.