1. Bernoulli Distribution

Use case: Single experiment with success/failure

PMF

$$ P(X = 1) = p, \quad P(X = 0) = 1 - p $$

Mean $\mathbb{E}[X] = p$ Variance $\mathrm{Var}(X) = p(1 - p)$ Special Properties

• Indicator random variable
• Building block of Binomial distribution

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2. Binomial Distribution

Use case: Number of successes in (n) independent Bernoulli trials

PMF

$$ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} $$

Mean $np$ Variance $np(1-p)$

Special Properties

• Sum of Bernoulli random variables
• Approximated by Poisson (small (p)) and Normal (large (n))

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3. Geometric Distribution

Use case: Number of trials until first success

PMF

$$ P(X = k) = (1-p)^{k-1}p, \quad k \ge 1 $$

Mean $\frac{1}{p}$ Variance $\frac{1-p}{p^2}$

Special Properties

• Memoryless: $$ P(X > s+t \mid X > s) = P(X > t) $$

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4. Poisson Distribution

Use case: Number of events in fixed time or space interval

PMF

$$ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$

Mean $\lambda$ Variance $\lambda$

Special Properties

• Independent increments
• limit of Binomial distribution

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5. Discrete Uniform Distribution

Use case: All outcomes equally likely

PMF
[
P(X = k) = \frac{1}{n}, \quad k = 1,2,\dots,n
]

Mean
[
\frac{n+1}{2}
]

Variance
[
\frac{n^2 - 1}{12}
]

---

🔹 Continuous Distributions

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6. Continuous Uniform Distribution

Use case: Random variable over an interval

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

CDF
[
F(x) = \frac{x-a}{b-a}
]

Mean
[
\frac{a+b}{2}
]

Variance
[
\frac{(b-a)^2}{12}
]

---

7. Exponential Distribution

Use case: Waiting time until next event

PDF

$$ f(x) = \lambda e^{-\lambda x}, \quad x \ge 0 $$

CDF

$$ F(x) = 1 - e^{-\lambda x} $$

Mean $\frac{1}{\lambda}$

Variance $\frac{1}{\lambda^2}$

Special Properties

• Memoryless (continuous analogue of Geometric)

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8. Normal (Gaussian) Distribution

Use case: Noise, natural variation, measurement errors

PDF $$

f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
$$

Mean $\mu$ Variance \sigma^2$

Special Properties

• Central Limit Theorem
• Fully determined by mean and variance
• Symmetric bell-shaped curve

---

9. Gamma Distribution

Use case: Waiting time until (k) events

PDF
[
f(x) = \frac{\lambda^k x^{k-1} e^{-\lambda x}}{\Gamma(k)}
]

Mean
[
\frac{k}{\lambda}
]

Variance
[
\frac{k}{\lambda^2}
]

Special Properties

• Generalizes Exponential distribution

---

10. Chi-Square Distribution

Use case: Sum of squared standard normals

Definition
[
X = \sum_{i=1}^k Z_i^2, \quad Z_i \sim \mathcal{N}(0,1)
]

Mean
[
k
]

Variance
[
2k
]

Special Properties

• Used in hypothesis testing

• Special case of Gamma distribution

---

11. Student’s t Distribution

Use case: Mean estimation with small samples

Mean
[
0 \quad (k > 1)
]

Variance
[
\frac{k}{k-2} \quad (k > 2)
]

Special Properties

• Heavier tails than Normal

• Converges to Normal as (k \to \infty)

---

12. Beta Distribution

Use case: Random variables on ([0,1]), Bayesian priors

PDF
[
f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}
]

Mean
[
\frac{\alpha}{\alpha + \beta}
]

Variance
[
\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}
]