1. **Binomial Distribution**: It models the number of successes in a fixed number of independent trials, each with the same probability of success $p$. The probability mass function (PMF) is given by $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $n$ is the number of trials and $k$ is the number of successes. 2. **Normal Distribution**: A continuous probability distribution characterized by its mean $\mu$ and variance $\sigma^2$. Its probability density function (PDF) is $$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$. It is symmetric and bell-shaped. 3. **Poisson Distribution**: Models the number of events occurring in a fixed interval of time or space when events occur independently at a constant rate $\lambda$. Its PMF is $$P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$$. 4. **Relations between Binomial, Normal, and Poisson Distributions**: - Binomial approximates to Normal when $n$ is large and $p$ is not too close to 0 or 1. - Binomial approximates to Poisson when $n$ is large and $p$ is small such that $np=\lambda$ is moderate. 5. **Uniform Distribution**: All outcomes in an interval $[a,b]$ are equally likely. The PDF is $$f(x) = \frac{1}{b-a}$$ for $a \leq x \leq b$. 6. **Exponential Distribution**: Models the time between events in a Poisson process with rate $\lambda$. The PDF is $$f(x) = \lambda e^{-\lambda x}$$ for $x \geq 0$. 7. **Correlation: Karl Pearson’s Coefficient**: Measures linear correlation between two variables $X$ and $Y$. It is $$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$$ where $\bar{x}$ and $\bar{y}$ are means. 8. **Rank Correlation**: Measures the relationship between rankings of two variables, e.g., Spearman’s rank correlation. 9. **Curve Fitting**: Finding a curve that best fits a set of data points, often by minimizing the sum of squared errors. 10. **Line of Regression**: The best-fit line predicting $Y$ from $X$ is $$Y = a + bX$$ where $$b = \frac{Cov(X,Y)}{Var(X)}$$ and $$a = \bar{Y} - b \bar{X}$$. --- ### Example Problems **1. Binomial Distribution** Problem: Find the probability of exactly 3 heads in 5 tosses of a fair coin. Solution: - $n=5$, $k=3$, $p=0.5$ - $$P(X=3) = \binom{5}{3} (0.5)^3 (0.5)^2 = 10 \times 0.125 \times 0.25 = 0.3125$$ **2. Normal Distribution** Problem: Find the probability that a normal variable with mean 0 and variance 1 lies between -1 and 1. Solution: - Use standard normal table or symmetry. - $$P(-1 < Z < 1) \approx 0.6826$$ **3. Poisson Distribution** Problem: Number of calls per hour is 3 on average. Find probability of exactly 5 calls. Solution: - $\lambda=3$, $k=5$ - $$P(X=5) = \frac{e^{-3} 3^5}{5!} = \frac{e^{-3} 243}{120} \approx 0.1008$$ **4. Uniform Distribution** Problem: Find the probability that a number chosen uniformly from [2,5] lies between 3 and 4. Solution: - $$P(3 < X < 4) = \frac{4-3}{5-2} = \frac{1}{3} \approx 0.3333$$ **5. Exponential Distribution** Problem: If the average time between arrivals is 2 minutes, find the probability that the next arrival is within 1 minute. Solution: - $\lambda = \frac{1}{2} = 0.5$ - $$P(X < 1) = 1 - e^{-0.5 \times 1} = 1 - e^{-0.5} \approx 0.3935$$ **6. Karl Pearson’s Correlation** Problem: Given data points $(1,2), (2,3), (3,5)$, find $r$. Solution: - Calculate means: $\bar{x}=2$, $\bar{y}=3.33$ - Calculate numerator and denominator: $$\sum (x_i - \bar{x})(y_i - \bar{y}) = (1-2)(2-3.33)+(2-2)(3-3.33)+(3-2)(5-3.33) = (-1)(-1.33)+0+1(1.67) = 1.33 + 0 + 1.67 = 3$$ $$\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2} = \sqrt{(1+0+1)(1.77+0.11+2.78)} = \sqrt{2 \times 4.66} = \sqrt{9.32} = 3.05$$ - $$r = \frac{3}{3.05} \approx 0.98$$ **7. Line of Regression** Problem: Find regression line of $Y$ on $X$ for above data. Solution: - $$b = \frac{Cov(X,Y)}{Var(X)} = \frac{3/3}{2/3} = \frac{1}{0.6667} = 1.5$$ - $$a = \bar{Y} - b \bar{X} = 3.33 - 1.5 \times 2 = 3.33 - 3 = 0.33$$ - Regression line: $$Y = 0.33 + 1.5 X$$