Chapter 20: Discrete Random Variables

Question 1

Define a random variable and state the types of random variables

Answer 1

RANDOM VARIABLE:
variable that uses numbers to describe the possible outcomes which could result from an experiment, and can be represented by X

• Discrete random variable: has a set of distinct possible values. determined by counting
• Continuous random variable: can take any value within some interval, determined by measuring

Question 2

Describe what is a discrete probability distribution and state the conditions for a valid probability distribution

Answer 2

• Probability that X takes on the value x = P(X=x)
• If X is a random variable with possible values {x₁, x₂, x₃…xₙ} and corresponding probabilities {p₁, p₂, p₃…pₙ}:

0≤pᵢ≤1
Σpᵢ (starting from i=1 to i=n) = 1

Question 3

Define a uniform discrete random variable

Answer 3

UNIFORM DISCRETE RANDOM VARIABLE:
when all possible values of X have the same probability (1/n) of occuring

Question 4

Find the mode and median from a probability distribution

Answer 4

MODE: most frequently occurring value of X (xᵢ when pᵢ is the highest)

MEDIAN: 50th percentile (xᵢ when p₁+ p₂+p₃+pₙ = 0.5)

Question 5

Find the expected value/mean value of a probability distribution

Answer 5

EXPECTED VALUE: E(X)

E(X) = Σxᵢpᵢ

Question 6

Determine if a game is fair

Answer 6

FAIR GAME: expected gain is 0 (when X denotes the player’s gains)

E(X) = 0

Question 7

Describe a binomial experiment and state the probability mass function of a binomial random variable (X) and its binomial distribution

Answer 7

BINOMIAL EXPERIMENT:
1) Fixed number of independent trials
2) Only 2 possible results per trial
3) Equal probability of success for each trial

P(X=x) = P(x) = (ⁿₓ) pˣ (1-p)ⁿ⁻ˣ
X∼B(n,p)

Question 8

Use GDC to find binomial probabilities and the mean and standard deviation of a binomial distribution

Answer 8

BINOMIAL PROBABILITY FUNCTION: P(x=k)
- Calculator page –> Menu, 5, 5, A

BINOMIAL CUMULATIVE PROBABILITY FUNCTION: P(x≤k) or P(x≥k)
- Calculator page –> Menu, 5, 5, B

MEAN (μ=np) AND STANDARD DEVIATION (σ=√np(1-p) ):
- Lists and Spreadsheets page –> key in all values in List A (name it x) –> menu, 4, 2, A in List B (name it z)
- Add Data and Statistics page –> enter variable x and z
- Add Calculator page –> menu, 6, 1, 1 (x₁ list: x; frequency list: z)
- Identify x̄ and σ