Mathematics | Continuous distributions

Welcome to Continuous Distributions: Understanding the Flow of Probability!

Hello future statistician! This chapter, Continuous Distributions, is where we transition from counting outcomes (Discrete Variables, like the number of heads) to measuring outcomes (Continuous Variables, like height, weight, or time).

Don't worry if the formulas involving integration seem intimidating. We will break them down step-by-step. Remember, integration is just finding the area under a curve, and in statistics, the area under the curve represents probability. You've got this!

What is a Continuous Random Variable (CRV)?

A Continuous Random Variable (CRV), \(X\), is a variable that can take any value within a given range.

• Examples: The time it takes a student to complete a test (e.g., 50.1 minutes, 50.1003 minutes, etc.), or the temperature of a room.
• Contrast with Discrete: Discrete variables jump from one integer value to the next (e.g., 1, 2, 3). Continuous variables flow smoothly.

Crucial Concept Alert! Probability at a Single Point

In continuous distributions, the probability of the variable taking exactly one specific value is always zero.

$$P(X = x) = 0$$

Why? Imagine trying to measure the exact height of a tree down to an infinite number of decimal places. The chance of hitting *that exact* infinite number is zero. Because of this, when dealing with continuous variables:

$$P(a \le X \le b) = P(a < X < b) = P(a \le X < b)$$

The endpoints (the equal signs) don't matter! This often simplifies calculations.

Section 1: The Probability Density Function (PDF)

The heart of any continuous distribution is its Probability Density Function (PDF), usually denoted by \(f(x)\).

Think of the PDF as the shape or recipe for your distribution. It describes how the probability is "distributed" across the range of possible values.

Key Properties of the PDF, \(f(x)\)

1. The Function must be Non-Negative

Since \(f(x)\) describes the likelihood of an outcome, the function itself cannot be negative.

$$f(x) \ge 0 \quad \text{for all } x$$

2. The Total Area Under the Curve must Equal 1

The total probability of all possible outcomes must be 1 (or 100%). In calculus terms, this means integrating the PDF over its entire defined range, \(R\), gives 1.

$$\int_R f(x) \, dx = 1$$

This property is often used to find unknown constants (like 'k') in a given PDF definition!

Calculating Probability Using the PDF

The probability that \(X\) falls between two values, \(a\) and \(b\), is the area under the curve of \(f(x)\) between those points.

$$P(a < X < b) = \int_a^b f(x) \, dx$$

Step-by-Step: Finding Probability

1. Identify the Limits: Determine the lower limit \(a\) and upper limit \(b\) for the probability you need.
2. Integrate: Calculate the definite integral of the PDF, \(f(x)\), between those limits.
3. Solve: Substitute the limits into the integrated function and find the final numerical value.

Quick Review: The PDF gives the *shape* of the distribution. Probability is always found by calculating the *area* (using integration).

Section 2: The Cumulative Distribution Function (CDF)

While the PDF tells you the density at a point, the Cumulative Distribution Function (CDF), \(F(x)\), tells you the probability accumulated up to that point \(x\).

Definition of the CDF, \(F(x)\)

The CDF is the probability that the random variable \(X\) takes a value less than or equal to a specific value \(x\).

$$F(x) = P(X \le x)$$

Relating PDF and CDF (Integration and Differentiation)

The relationship between \(f(x)\) and \(F(x)\) is the fundamental theorem of calculus:

1. To go from PDF to CDF: Integrate $$F(x) = \int_{-\infty}^{x} f(t) \, dt$$ (Note: In practice, we integrate from the lowest boundary where \(f(x)\) is defined, say \(a\), up to \(x\).)
2. To go from CDF to PDF: Differentiate $$f(x) = F'(x) = \frac{d}{dx} F(x)$$ This is a great trick for verifying your calculations!

Properties of the CDF

1. Starting Point: The CDF starts at 0. If the minimum value is \(a\), then \(F(a) = 0\).
2. Ending Point: The CDF ends at 1. If the maximum value is \(b\), then \(F(b) = 1\).
3. Non-Decreasing: As \(x\) increases, \(F(x)\) must never decrease (accumulated probability can only increase or stay the same).

Calculating Probability Using the CDF

If you already have the CDF, calculating probabilities is much simpler than integration!

$$P(a < X < b) = F(b) - F(a)$$

Analogy: If \(F(b)\) is the total weight of flour in the bag up to point \(b\), and \(F(a)\) is the weight up to point \(a\), then \(F(b) - F(a)\) is the weight of flour *between* \(a\) and \(b\).

Common Mistake to Avoid

When calculating the CDF, \(F(x)\), make sure you include the constant of integration, \(C\). However, since we define the lower limit of integration as the minimum value of \(X\), we usually find \(C=0\).

Always check: If \(X\) is defined for \(x \ge a\), then setting \(F(a) = 0\) helps you find \(C\).

Section 3: Expectation, Variance, and Median

Expectation (The Mean)

The Expectation or Mean, \(E(X)\) or \(\mu\), is the long-run average value of the random variable. It is the balance point of the distribution.

Just like in discrete variables where we calculate \(\sum x P(X=x)\), here we replace the sum with an integral and the probability \(P(X=x)\) with the density function \(f(x)\).

$$E(X) = \mu = \int x f(x) \, dx$$

Expectation of a Function of X

If you need to find the expectation of a function of \(X\), say \(g(X)\):

$$E(g(X)) = \int g(x) f(x) \, dx$$

The most important case is finding \(E(X^2)\) by setting \(g(x) = x^2\):

$$E(X^2) = \int x^2 f(x) \, dx$$

Variance

The Variance, \(\text{Var}(X)\), measures the spread or dispersion of the distribution around the mean.

The formula is exactly the same as for discrete variables:

$$\text{Var}(X) = E(X^2) - [E(X)]^2$$

The Standard Deviation is \(\sigma = \sqrt{\text{Var}(X)}\).

The Median (m)

The Median, \(m\), is the value that splits the distribution into two equal halves. Half the probability mass lies below it, and half lies above it.

Therefore, the median \(m\) is the value such that:

$$P(X \le m) = 0.5$$

You can find the median by solving either of these equations:

$$\int_{-\infty}^{m} f(x) \, dx = 0.5 \quad \text{OR} \quad F(m) = 0.5$$

Tip: It is almost always easier to use the CDF method, \(F(m) = 0.5\), if you have already calculated \(F(x)\).

Section 4: The Continuous Uniform Distribution (The Rectangle)

The Continuous Uniform Distribution is the simplest type of continuous distribution. It assumes that the random variable is equally likely to take any value within a defined interval, but cannot exist outside that interval.

We denote it as: $$X \sim U(a, b)$$ where \(a\) is the minimum value and \(b\) is the maximum value.

The PDF of U(a, b)

Since the probability is spread out "uniformly" (evenly), the PDF looks like a rectangle.

The height of the rectangle, \(f(x)\), must ensure the total area is 1. The width is \((b-a)\).

$$f(x) = \frac{1}{\text{width}} = \frac{1}{b-a} \quad \text{for } a \le x \le b$$ $$f(x) = 0 \quad \text{otherwise}$$

Calculating Probability in a Uniform Distribution

For uniform distributions, you do not need integration for probabilities! Since the shape is a rectangle, probability is just:

$$P(x_1 < X < x_2) = \text{Height} \times \text{Width}$$ $$P(x_1 < X < x_2) = \left(\frac{1}{b-a}\right) \times (x_2 - x_1)$$

The CDF of U(a, b)

The CDF is an increasing straight line between \(a\) and \(b\).

• For \(x < a\), \(F(x) = 0\)
• For \(a \le x \le b\), $$F(x) = \frac{x-a}{b-a}$$
• For \(x > b\), \(F(x) = 1\)

Quick Formulas for Mean and Variance

One of the major advantages of recognizing a uniform distribution is that you can use these quick formulas instead of performing the complex integration of \(xf(x)\) and \(x^2f(x)\).

Mean (Expectation)

The mean is simply the midpoint of the interval.

$$E(X) = \mu = \frac{a+b}{2}$$

Variance

$$\text{Var}(X) = \frac{(b-a)^2}{12}$$

Memory Aid: The variance formula involves 12 because 12 is the smallest integer divisible by both 3 and 4, which come up during the integration process when calculating \(E(X)\) and \(E(X^2)\) for the uniform case. Just remember that lovely number 12!

Final Review and Key Takeaways

Checklist for Continuous Distribution Problems

1. Identify the Function: Is it a general PDF (requires integration) or a Uniform distribution (can use quick area/formula)?
2. Total Area Check: Always confirm that \(\int f(x) \, dx = 1\). If there is a constant (k), find it first!
3. Probability: If using PDF, integrate \(f(x)\). If using CDF, subtract \(F(b) - F(a)\).
4. Mean/Variance: Remember to use the formula \(E(X) = \int x f(x) \, dx\) and then \(\text{Var}(X) = E(X^2) - [E(X)]^2\).

Did you know? The Normal Distribution, which you might study later, is also a continuous distribution. It’s arguably the most famous curve in statistics! Its PDF is incredibly complex, which is why we rely on tables or calculators to find its probabilities, rather than integration.

Keep practicing your integration skills! In Statistics 2, the math is often just applied calculus. The concepts are logical; it’s the execution that requires practice. You are mastering a complex link between algebra, calculus, and real-world statistics. Great job!

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