Discrete R.V

Overview: This video covers probability distributions and discrete random variables, building on prior discussions about experiments and sample spaces.
- Sample Space: The set of all possible outcomes of an experiment.

Definition: A random variable is a numerical value assigned to each outcome in a sample space.
- For example, in the experiment of rolling a die:
- Let random variable X represent the number rolled.
  - Possible values for X: {1, 2, 3, 4, 5, 6}.
- An alternative random variable Y could represent parity:
  - Y = 0 for even numbers, Y = 1 for odd numbers.
Another Example: Selecting a policeman randomly from a local police department.
- Define random variable X as the number of sick days taken by the policeman last year.
- Possible values for X: {0, 1, 2, …, 10} (assuming a maximum of 10 sick days).
- Define random variable Y as the number of arrests made by the policeman throughout their career.
- Possible values for Y: {0, 1, 2, …} (theoretically up to infinity).
- A categorical variable could also be defined:
- Z = 0 if the policeman is under 40 years old, Z = 1 if they are 40 or older.

Probability Distribution Function (PDF): A table or equation that explains the probabilities of each possible value of a random variable.
- For example, considering rolling a die:
- Each value of X can take on probabilities:
  - P(X=1) = $rac{1}{6}$ , P(X=2) = $rac{1}{6}$ , and so forth for each die face.
Requirements of PDF:
- Each probability must be between 0 and 1.
- The sum of all probabilities must equal 1.

Experiment Definition: Airlines often overbook flights.
- Define random variable X as the number of ticketed passengers who show up for a flight with 50 seats and 55 passengers.
- PDF of X fluctuates between 45 to 55 for the number of passengers showing up.
- Requirements: Total probabilities from 45 to 55 passengers must sum to 1 (indicating all possibilities considered).

Specific Probability Example: To find the probability that exactly 50 people show up:
- Notation: P(X=50) from the PDF = 17% chance.
To find the probability that more than 50 people show up:
- P(X > 50) = P(X=51) + P(X=52) + P(X=53) + P(X=54) + P(X=55).
- This also happens to equal 17% (for the sake of illustration).

Population Mean (μ): Represents the average number of passengers showing up for the flight, denoted as $ext{M}$ .
- Formula for calculating the mean:
  $ext{μ} = ext{sum of (each value of X)} imes ext{P(X)}$ .
- Calculate average number: Multiply values of X by their respective probabilities and sum.
Example Mean: Average of passengers showing up noted as 48.8.
- Important: An average of 48.8 does not imply that such a number of passengers physically shows up; rather it indicates a statistical average.
Standard Deviation (σ): Measures how spread out the values are around the mean.
- Denoted as $ext{σ}$ , calculated using:
  $ext{σ} = ext{sqrt}\bigg( ext{sum of}((X - μ)^{2} imes P(X))\bigg)$ .
- Describes the range of values compared to the mean.

Setup: Calculate importance of spread around the mean.
- Setup columns for computations:
- One for X - μ
- Another for (X - μ)²
- Final column for (X - μ)² × P(X)
Final Output: To find the standard deviation,
- Take $ext{σ} = ext{sqrt}\bigg( ext{sum of all values in the final column}\bigg)$ .

Review computation and application of discrete random variables, probability distributions, mean, and standard deviation.
Importance: Understanding these concepts creates a foundation for further study in statistics and probability theory.