1. **Measures of Central Tendency: Mean, Mode, Median** - Mean is the average: $$\text{Mean} = \frac{\sum x_i}{n}$$ where $x_i$ are data points and $n$ is the number of points. - Mode is the most frequent value. - Median is the middle value when data is ordered. 2. **Measures of Dispersion** - Dispersion shows data spread: range, variance, standard deviation. - Range = max value - min value. 3. **Variance and Standard Deviation** - Variance: $$\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$$ where $\mu$ is mean. - Standard deviation: $$\sigma = \sqrt{\sigma^2}$$ 4. **Definition of Probability** - Probability of event $A$: $$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$$ 5. **Laws of Probability** - Addition rule: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ - Multiplication rule for independent events: $$P(A \cap B) = P(A) \times P(B)$$ 6. **Expectation, Variance, and SD of Random Variables** - Expectation (mean): $$E(X) = \sum x_i P(x_i)$$ - Variance: $$Var(X) = E[(X - E(X))^2] = \sum (x_i - E(X))^2 P(x_i)$$ - SD: $$\sqrt{Var(X)}$$ 7. **Calculations with Discrete and Continuous Random Variables** - Discrete: sum over values. - Continuous: use integrals for expectation and variance. 8. **Types of Distributions** 8.1 Binomial Distribution: - $$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$ - Mean: $$np$$, Variance: $$np(1-p)$$ 8.2 Poisson Distribution: - $$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$$ - Mean and Variance: $$\lambda$$ 8.3 Normal Distribution: - PDF: $$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$ - Mean: $$\mu$$, Variance: $$\sigma^2$$ 9. **Mean, Variance, and SD of Probability Distributions** - Use formulas from step 6 for any distribution. 10. **Application of Probability Distributions** - Example: Binomial with $n=5$, $p=0.4$, find $P(X=2)$: $$P(X=2) = \binom{5}{2} (0.4)^2 (0.6)^3 = 10 \times 0.16 \times 0.216 = 0.3456$$ - Use graphs to visualize distributions (not shown here). This summary covers key concepts and formulas for statistics and probability calculations.