Standard normal distribution

Question 1

Probability density function - 3

Answer 1

• Curve p(x)
• p(x) ≥ 0 for every x
• The total area under the curve is 1
• The probability that X takes a value between a and b

Question 2

Normal distribution - 3 + 3

Answer 2

• Probability density is:
• • Continuous
• • Bell shaped
• • Symmetric
• Mean of X is μ (miu)
• Standard deviation is σ (sigma)

Question 3

Normal distribution formula - 1

Answer 3

• p(x) = 1 / σ(√2π) * e^(-(1/2) * ( (x-μ) /σ )^2

Question 4

Normal distribution distribution - 3

Answer 4

• 68% of probability mass -> (μ - σ and μ + σ)
• 95% of probability mass -> (μ - 2σ and μ + 2σ)
• 99.7% of probability mass -> (μ − 3σ and μ + 3σ)

Question 5

Determine probabilities of normal distribution - 3

Answer 5

• P(X ≤ z) = cumulative area under the density curve to the left of z.
• P(X∈ [a, b]) = P(X ≤ b) - P(X ≤ a)
• P(X ≥ b) = 1 - P(X ≤ b)

Question 6

Normal distribution notation - 2

Answer 6

• N(μ,σ^2)

- Standard normal distribution N(0,1)

Question 7

pnorm and tables - 2

Answer 7

• pnorm(x) gives probability P(X ≤ x)

- Tables can show the cumulative area under the curve to the left of a z score P(X ≤ z)