Statistical distributions - Topic 4 Flashcards
Understand simple & discrete distributions, normal distribution, and appopriate probability distribution
Year 1 - Chapter 6.1
What does a probability distribution describe?
The probability of any outcome in a sample space, the range of values that a random variable can take.
Year 1 - Chapter 6.1
What is the probability mass function of a fair dice roll?
P(X=x) = 1/6, x = 1, 2, 3, 4, 5, 6
The sum of all probabilities of all outcomes of an event add up to 1. For a random variable X, you can write ΣP(X=x) = 1
Year 1 - Chapter 6.2
When can you model X binomially?
When:
- there are a fixed number of trials, n
- there are two possible outcomes
- there is a fixed probability of success, p
- the trials are independent of each other
Year 1 - Chapter 6.2
What is the probability mass function of binomial distribution?
ⁿCᵣ x pʳ x (1-p)ⁿ⁻ʳ
ⁿCᵣ = n!/(r!(n-r)!)
Year 1 - Chapter 6.3
The random variable X ~ B(20, 0.4) Find:
a. P(X ≤ 7)
b. P(X < 6)
c. P(X ≥ 15)
a. = 0.416
b. = P(X ≤ 5) = 0.126
c. = 1 - P(X ≤ 14) = 0.00161
Year 2 - Chapter 3.1
What features does normal distribution have?
- Has parameters μ, the population mean and σ², the population variance
- Is symmetrical (mean = median = mode)
- Has a bell-shaped curve with asymptotes at each end
- Has a total area under the curve equal to 1
- Has points of inflection at μ+σ and μ-σ
Year 2 - Chapter 3.4
What is the standard normal distribution?
X ~ N(μ, σ²)
Codification of X; Z = (X - μ)/σ
Year 2 - Chapter 3.6
When are you able to approximate binomial distribution?
If n is large and p is close to 0.5, then the binomial distribution X ~ B(n, p) can be approximated by the normal distribution N(μ, σ²) where:
- μ = np
- σ = √(np(1-p))