1. Continuous Probability Distributions

Continuous probability distributions describe random variables that can take any value in an interval. They are represented using a Probability Density Function (PDF).

2. Uniform Distribution (Continuous)

Definition: A continuous random variable $X$ is uniformly distributed over $a, b$ if all values in the interval are equally likely.

• Random variable: $X \sim U(a,b)$

• PDF:

  $$f(x) = \frac{1}{b-a}, \quad a \le x \le b$$

• CDF (Cumulative Distribution Function):

  $$F(x) = P(X \le x) = \frac{x-a}{b-a}, \quad a \le x \le b$$

• Mean and Variance:

  $$E(X) = \frac{a+b}{2}, \quad Var(X) = \frac{(b-a)^2}{12}$$

Example:

• A bus arrives at a stop uniformly between 8:00 and 8:30 → $X \sim U(0,30)$ minutes.

Graph: Rectangular shape (height = $1/(b-a)$).

3. Exponential Distribution

Definition: Exponential distribution models time between events in a Poisson process.

• Random variable: $X \ge 0$, time until next event
• Parameter: $\lambda > 0$, the rate of occurrence

PDF:

CDF:

Mean and Variance:

Example:

• Time between arrivals of customers at a store (Poisson process).

Graph: Decreasing exponential curve starting at $\lambda$ at $x=0$.

4. Normal Distribution

Definition: Normal distribution, also called Gaussian distribution, is the most common bell-shaped continuous distribution.

• Random variable: $X \sim N(\mu, \sigma^2)$

• Parameters:

  • $\mu$ = mean (center)
  • $\sigma^2$ = variance (spread)

PDF:

Properties:

1. Symmetric about $\mu$
2. Mean = Median = Mode = $\mu$
3. About 68% of data within $\mu \pm \sigma$, 95% within $\mu \pm 2\sigma$, 99.7% within $\mu \pm 3\sigma$
4. Total area under curve = 1

Example:

• Heights of adults, IQ scores, measurement errors.

Graph: Bell-shaped curve, peak at $\mu$, spread determined by $\sigma$.

5. Summary Table

Key Points to Remember

1. Uniform → all outcomes equally likely.
2. Exponential → models waiting times, memoryless.
3. Normal → bell-shaped, widely used in statistics, many natural phenomena.