Probability Distributions: Binomial and Normal

Discrete Random Variable with Binary Outcomes: For such variables, the binomial model (or binomial probability distribution) is used.
pbinom vs. dbinom in R:
- pbinom: Used to find the probability of a range of values for a discrete random variable (e.g., $X \le 5$ successes).
- dbinom: Used to find the probability of exactly one specific value for a discrete random variable (e.g., $X = 13$ successes).
Airline Example (Binomial Model):
- Scenario: Probability of a passenger no-showing is $9\%$ ( $p = 0.09$ ), and $140$ tickets ( $n = 140$ ) were sold.
- Expected value (most likely number of no-shows): $n \times p = 140 \times 0.09 = 12.6$ , rounded to $13$ no-shows (since people are whole numbers).
- To find the probability of exactly $13$ no-shows, dbinom would be used.
Binomial Random Variable Criteria:
- Binary outcomes: Only two possible outcomes (e.g., left-handed vs. not left-handed, renew vs. not renew). Even if a categorical variable has multiple options, it can often be reframed into a binary outcome (e.g.,