1. **Stating the problem:** We want to understand what random variables are, their types, and see examples with solutions. 2. **Definition:** A random variable is a function that assigns a numerical value to each outcome in a sample space of a random experiment. 3. **Types of random variables:** - **Discrete random variable:** Takes countable values (e.g., number of heads in 3 coin tosses). - **Continuous random variable:** Takes values in a continuous range (e.g., height of a person). 4. **Example 1 (Discrete):** Suppose we toss a fair coin 3 times. Let $X$ be the number of heads. - Possible values of $X$ are $0,1,2,3$. - Probability mass function (pmf): $$P(X=k) = \binom{3}{k} \left(\frac{1}{2}\right)^k \left(\frac{1}{2}\right)^{3-k} = \binom{3}{k} \left(\frac{1}{2}\right)^3$$ - For example, $P(X=2) = \binom{3}{2} \left(\frac{1}{2}\right)^3 = 3 \times \frac{1}{8} = \frac{3}{8}$. 5. **Example 2 (Continuous):** Let $Y$ be the height of a randomly selected person from a population, modeled as a continuous random variable with probability density function (pdf) $f(y)$. - For example, if $Y$ is uniformly distributed between 150 cm and 190 cm, then $$f(y) = \frac{1}{190-150} = \frac{1}{40} \quad \text{for } 150 \leq y \leq 190$$ - Probability that height is between 160 and 170 cm: $$P(160 \leq Y \leq 170) = \int_{160}^{170} \frac{1}{40} dy = \frac{170-160}{40} = \frac{10}{40} = \frac{1}{4}$$ 6. **Summary:** Random variables map outcomes to numbers. Discrete variables have countable values with pmf, continuous variables have ranges with pdf. Final answers: - Discrete example: $P(X=2) = \frac{3}{8}$ - Continuous example: $P(160 \leq Y \leq 170) = \frac{1}{4}$