1. **Discrete and Continuous Random Variables:** A discrete random variable takes countable values (e.g., number of heads in coin tosses). A continuous random variable takes values in an interval (e.g., height of people). 2. **Joint Distribution:** Describes the probability distribution of two or more random variables simultaneously. For discrete variables, it's a joint probability mass function $P(X=x, Y=y)$. For continuous, it's a joint probability density function $f_{X,Y}(x,y)$. 3. **Probability Distribution Function:** For discrete variables, the probability mass function (pmf) $P(X=x)$ gives probabilities. For continuous variables, the probability density function (pdf) $f_X(x)$ satisfies $P(a \leq X \leq b) = \int_a^b f_X(x) dx$. 4. **Conditional Distribution:** The distribution of one variable given another. For discrete: $P(X=x|Y=y) = \frac{P(X=x,Y=y)}{P(Y=y)}$. For continuous: $f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}$. 5. **Mathematical Expectations:** The expected value or mean $E[X]$ is the average value of $X$. For discrete: $E[X] = \sum_x x P(X=x)$. For continuous: $E[X] = \int x f_X(x) dx$. 6. **Moments:** The $n$th moment about zero is $E[X^n]$. The central moment is $E[(X - E[X])^n]$. 7. **Moment Generating Functions (MGF):** Defined as $M_X(t) = E[e^{tX}]$. MGFs uniquely determine the distribution and help find moments by differentiation. 8. **Variance and Correlation Coefficients:** Variance measures spread: $Var(X) = E[(X - E[X])^2]$. Correlation coefficient between $X$ and $Y$ is $\rho_{X,Y} = \frac{Cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$ where $Cov(X,Y) = E[(X - E[X])(Y - E[Y])]$. 9. **Chebyshev’s Inequality:** For any random variable $X$ with mean $\mu$ and variance $\sigma^2$, and any $k > 0$: $$P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}$$ This bounds the probability that $X$ deviates from its mean. 10. **Skewness and Kurtosis:** Skewness measures asymmetry: $\gamma_1 = \frac{E[(X - \mu)^3]}{\sigma^3}$. Kurtosis measures tail heaviness: $\gamma_2 = \frac{E[(X - \mu)^4]}{\sigma^4}$. --- **Example Problems:** 1. Discrete RV: Let $X$ be number of heads in 3 coin tosses. Find $P(X=2)$. 2. Continuous RV: Let $X$ have pdf $f_X(x) = 2x$ for $0 \leq x \leq 1$. Find $E[X]$. 3. Joint Distribution: Given joint pmf $P(X=0,Y=0)=0.1$, $P(X=0,Y=1)=0.4$, $P(X=1,Y=0)=0.2$, $P(X=1,Y=1)=0.3$, find $P(X=1|Y=1)$. 4. MGF: For $X$ with $P(X=1)=0.5$, $P(X=-1)=0.5$, find $M_X(t)$. 5. Variance: For $X$ above, find $Var(X)$. 6. Chebyshev: For $X$ with mean 10 and variance 4, find upper bound for $P(|X-10| \geq 6)$. 7. Skewness: For $X$ with pdf symmetric about mean, what is skewness? --- **Solutions:** 1. $P(X=2) = \binom{3}{2} (0.5)^2 (0.5)^1 = 3 \times 0.25 \times 0.5 = 0.375$ 2. $E[X] = \int_0^1 x \cdot 2x dx = 2 \int_0^1 x^2 dx = 2 \times \frac{1}{3} = \frac{2}{3}$ 3. $P(X=1|Y=1) = \frac{P(X=1,Y=1)}{P(Y=1)} = \frac{0.3}{0.4+0.3} = \frac{0.3}{0.7} \approx 0.429$ 4. $M_X(t) = E[e^{tX}] = 0.5 e^{t} + 0.5 e^{-t} = \cosh(t)$ 5. $E[X] = 0$, $E[X^2] = 0.5(1)^2 + 0.5(-1)^2 = 1$, so $Var(X) = 1 - 0^2 = 1$ 6. By Chebyshev: $P(|X-10| \geq 6) \leq \frac{4}{6^2} = \frac{4}{36} = \frac{1}{9} \approx 0.111$ 7. Skewness of symmetric distribution is 0.