Unit-4: Special Continuous Distributions

1. Normal Distribution (Gaussian Distribution)

The Normal Distribution is a symmetric, bell-shaped distribution characterized by its mean (μ) and variance (σ²).

Probability Density Function (PDF):

• μ: Mean (location parameter).
• σ: Standard deviation (scale parameter).
• σ²: Variance.

Graph of Normal Distribution (Bell Curve):

Cumulative Distribution Function (CDF):

• erf(z): Error function.

Key Properties:

• Symmetric about the mean μ.
• Mean = Median = Mode = μ.
• 68% of data lies within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.

Moment Generating Function (MGF):

Standard Deviation

Standard deviation ($\sigma$) is a measure of the dispersion or spread of data points around the mean ($\mu$) in a distribution. It is the square root of the variance (${\sigma }^2$).

Formula

Population Standard Deviation:

Sample Standard Deviation:

Graphical Representation (Normal Distribution with Different σ):

Key Properties

• Low σ: Data points are close to the mean (less spread).
• High σ: Data points are far from the mean (more spread).
• Unit: Same as the original data.
• Non-negative: $\sigma \ge 0$.

Empirical Rule (For Normal Distributions)

• $\mu \pm \sigma$ covers ~68% of data.
• $\mu \pm 2\sigma$ covers ~95% of data.
• $\mu \pm 3\sigma$ covers ~99.7% of data.

Example Calculation

For dataset: [2, 4, 4, 4, 5, 5, 7, 9]

1. Mean ($\overline{x}$) = 5
2. Squared deviations: [9, 1, 1, 1, 0, 0, 4, 16]
3. Variance = 32 / 7 ≈ 4.57
4. Standard deviation = √4.57 ≈ 2.14

2. Gamma Distribution

The Gamma Distribution is a two-parameter family of continuous distributions, often used to model waiting times or lifetimes.

Probability Density Function (PDF):

• α: Shape parameter (α > 0).
• β: Rate parameter (β > 0).
• Γ(α): Gamma function, defined as Γ(α) = ∫₀^∞ t^α-1 e^-t dt.

Graph of Gamma Distribution:

Cumulative Distribution Function (CDF):

• γ(α, βx): Lower incomplete gamma function.

Key Properties:

• If α = 1, it reduces to the Exponential distribution.
• $\mathrm{Mean\; =}\frac{\alpha }{\beta }$
• $\mathrm{Variance\; =}\frac{\alpha }{\mathrm{\beta ²}}$.

Moment Generating Function (MGF):

3. Exponential Distribution

The Exponential Distribution is a special case of the Gamma Distribution (when α = 1). It models the time between events in a Poisson process.

Probability Density Function (PDF):

• λ: Rate parameter (λ > 0).

Graph of Exponential Distribution:

Cumulative Distribution Function (CDF):

Key Properties:

• Memoryless property: P(X > s + t | X > s) = P(X > t).
• $\mathrm{Mean\; =}\frac{1}{\lambda }$
• $\mathrm{Variance\; =}\frac{1}{\mathrm{\lambda ²}}$.

Moment Generating Function (MGF):

4. Moment Generating Functions (MGFs)

The Moment Generating Function (MGF) of a random variable X is defined as:

• E[·]: Expectation operator.

MGFs of the Distributions:

Distribution	MGF
Normal	MX(t) = eμt + (1/2)σ²t²
Gamma	MX(t) = (1 - t/β)-α, for t < β
Exponential	MX(t) = λ / (λ - t), for t < λ

MCQ Questions