Statistical distributions - Topic 4

Question 1

Year 1 - Chapter 6.1

What is the probability mass function of a fair dice roll?

Answer 1

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

Question 2

Year 1 - Chapter 6.1

What does a probability distribution describe?

Answer 2

The probability of any outcome in a sample space, the range of values that a random variable can take.

Question 3

Year 1 - Chapter 6.2

What is the probability mass function of binomial distribution?

Answer 3

ⁿCᵣ x pʳ x (1-p)ⁿ⁻ʳ

ⁿCᵣ = n!/(r!(n-r)!)

Question 4

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)

Answer 4

a. = 0.416
b. = P(X ≤ 5) = 0.126
c. = 1 - P(X ≤ 14) = 0.00161

Question 5

Year 2 - Chapter 3.1

What features does normal distribution have?

Answer 5

• 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

Question 6

Year 2 - Chapter 3.4

What is the standard normal distribution?

Answer 6

X ~ N(μ, σ²)

Codification of X; Z = (X - μ)/σ

Question 7

How do you find μ and σ?

Example:

Answer 7

Z = (X - μ)/σ

X ~ N(μ, 3²), P(X > 20) = 0.2
P(Z > (20 - μ)/3) = 0.2
Use the z-values table to find out z; z = 0.8416
0.8416 = (20 - μ)/3
μ = 17.4752 = 17.5

Question 8

Year 2 - Chapter 3.6

When are you able to approximate binomial distribution?

Answer 8

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))

If you are using a normal approximation to a binomial distribution, you need to apply a continuity correction when calculating probabilities.