In probability and statistics, distributions form the backbone of how we understand randomness and uncertainty. After studying discrete distributions, the next important step is learning about continuous distributions.
Continuous distributions are crucial in real-world applications: from modeling exam scores, waiting times, and product lifespans to financial returns, risk analysis, and even machine learning algorithms.
This post will walk you through the definition, key types, solved examples, and exercises related to continuous distributions.
🔹 What is a Continuous Distribution?
A continuous distribution applies when a random variable can take any real value within an interval (not just whole numbers, like discrete variables).
Instead of assigning probabilities to exact values (like in discrete distributions), continuous random variables use a probability density function (PDF).
Formally, P(a≤X≤b)=∫abf(x) dxP(a \leq X \leq b) = \int_a^b f(x)\,dxP(a≤X≤b)=∫abf(x)dx
where:
- $f(x) \geq 0$ for all $x$ (density is never negative)
- $\int_{-\infty}^{\infty} f(x),dx = 1$ (total probability = 1)
💡 Important Note: For continuous random variables, P(X=a)=0P(X = a) = 0P(X=a)=0
since probability is spread smoothly over an interval.
🔹 Key Continuous Distributions
1. Uniform Distribution
- All values in an interval $[a,b]$ are equally likely.
f(x)={1b−a,a≤x≤b0,otherwisef(x) = \begin{cases} \frac{1}{b-a}, & a \leq x \leq b \\ 0, & \text{otherwise} \end{cases}f(x)={b−a1,0,a≤x≤botherwise
- Mean: $E[X] = \frac{a+b}{2}$
- Variance: $\frac{(b-a)^2}{12}$
💡 Example: Waiting time for a bus that arrives uniformly between 0 and 10 minutes.
2. Normal (Gaussian) Distribution
- The most famous distribution, often called the “bell curve.”
- Denoted as $X \sim N(\mu, \sigma^2)$.
f(x)=12πσ2exp(−(x−μ)22σ2)f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)f(x)=2πσ21exp(−2σ2(x−μ)2)
- Mean: $\mu$
- Variance: $\sigma^2$
- Symmetric, bell-shaped, and widely used due to the Central Limit Theorem.
💡 Example: Human heights, IQ scores, and measurement errors often follow a normal distribution.
3. Exponential Distribution
- Models the time between events occurring at a constant rate $\lambda$.
f(x)={λe−λx,x≥00,x<0f(x) = \begin{cases} \lambda e^{-\lambda x}, & x \geq 0 \\ 0, & x < 0 \end{cases}f(x)={λe−λx,0,x≥0x<0
- Mean: $\frac{1}{\lambda}$
- Variance: $\frac{1}{\lambda^2}$
💡 Example: Time between customer arrivals at a bank counter.
4. Gamma Distribution
- A generalization of the exponential distribution, often used in queuing theory and reliability analysis.
f(x)=λkΓ(k)xk−1e−λx,x≥0f(x) = \frac{\lambda^k}{\Gamma(k)} x^{k-1} e^{-\lambda x}, \quad x \geq 0f(x)=Γ(k)λkxk−1e−λx,x≥0
- Mean: $\frac{k}{\lambda}$
- Variance: $\frac{k}{\lambda^2}$
5. Beta Distribution
- Defined on the interval $[0,1]$ with two shape parameters $\alpha, \beta$.
f(x)=Γ(α+β)Γ(α)Γ(β)xα−1(1−x)β−1,0<x<1f(x) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1} (1-x)^{\beta-1}, \quad 0<x<1f(x)=Γ(α)Γ(β)Γ(α+β)xα−1(1−x)β−1,0<x<1
- Mean: $\frac{\alpha}{\alpha+\beta}$
- Variance: $\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$
💡 Example: Used in Bayesian statistics, modeling probabilities and proportions.
🔹 Worked Examples
✅ Example 1: Uniform Distribution
A bus arrives uniformly between 0 and 20 minutes. What is the probability that you wait less than 5 minutes?
- Here, $a=0$, $b=20$, so $f(x) = 1/20$.
P(X<5)=∫05120 dx=520=0.25P(X<5) = \int_0^5 \frac{1}{20}\,dx = \frac{5}{20} = 0.25P(X<5)=∫05201dx=205=0.25
👉 Answer: 25% chance of waiting less than 5 minutes.
✅ Example 2: Exponential Distribution
The lifetime of a light bulb follows an exponential distribution with $\lambda = 0.2$. What’s the probability the bulb lasts more than 10 hours? P(X>10)=∫10∞0.2e−0.2x dx=e−2≈0.135P(X>10) = \int_{10}^{\infty} 0.2 e^{-0.2x}\,dx = e^{-2} \approx 0.135P(X>10)=∫10∞0.2e−0.2xdx=e−2≈0.135
👉 Answer: 13.5% chance.
✅ Example 3: Normal Distribution
Heights of students are normally distributed with mean $\mu=170$ cm and standard deviation $\sigma=10$ cm. What is the probability a student is taller than 180 cm? Z=X−μσ=180−17010=1Z = \frac{X-\mu}{\sigma} = \frac{180-170}{10} = 1Z=σX−μ=10180−170=1
From Z-tables, $P(Z>1) = 0.1587$.
👉 Answer: About 15.9% of students are taller than 180 cm.
🔹 Practice Exercises
Try solving these problems:
- A random variable is uniformly distributed on $[2,6]$. Find $P(X>4)$.
- If the lifetime of a machine follows an exponential distribution with mean 5 years, what’s the probability it lasts at least 10 years?
- Suppose $X \sim N(50, 9)$. Find $P(47 \leq X \leq 53)$.
- A Gamma distribution with parameters $k=3, \lambda=2$ is given. Find its mean and variance.
- For a Beta distribution with $\alpha=2, \beta=3$, compute the mean and variance.
🔹 Summary
- Continuous distributions model variables that can take infinitely many values within an interval.
- Probabilities are found using integrals of the PDF.
- Key distributions include:
- Uniform (equal likelihood)
- Normal (bell-shaped, most common)
- Exponential (time between events)
- Gamma (generalized exponential)
- Beta (probabilities and proportions)
- The Normal distribution is particularly powerful due to its role in the Central Limit Theorem.
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