2.13.1 Normal Flashcards
(15 cards)
What is a normal distribution?
A continuous probability distribution that follows a bell-shaped, symmetric curve centered around the mean.
What are the two parameters of a normal distribution?
- Mean (μ): The center of the distribution.
- Standard deviation (σ): The measure of spread.
How do we denote that a random variable ( X ) follows a normal distribution?
X ∼ N(μ, σ²)
where ( μ ) is the mean and ( σ² ) is the variance.
What does the PDF of a normal distribution look like?
A bell-shaped curve that is symmetric around the mean ( μ ).
What happens to the normal distribution curve when ( σ ) increases or decreases?
- Larger ( σ ): The curve spreads out (wider).
- Smaller ( σ ): The curve becomes narrower and taller.
What is a standard normal distribution?
A normal distribution where:
- The mean is 0: ( μ = 0 )
- The standard deviation is 1: ( σ = 1 )
It is denoted as ( Z ∼ N(0,1) ).
What is the formula to convert a normal variable ( X ) into a standard normal variable ( Z )?
Z = (X - μ) / σ
This transformation allows us to use the standard normal table to find probabilities.
Why do we use a normal table (Z-table)?
The CDF of a normal distribution does not have a closed formula, so we use the Z-table to look up cumulative probabilities.
How do you find ( P(X ≤ x) ) for a normal distribution?
Convert ( X ) to a Z-score, then use the Z-table to find the cumulative probability.
How do you find ( P(X > x) )?
P(X > x) = 1 - P(X ≤ x)
Look up ( P(X ≤ x) ) in the Z-table and subtract from 1.
What is the empirical rule (68-95-99.7 rule)?
For a normal distribution:
- 68% of values are within 1σ of the mean.
- 95% of values are within 2σ of the mean.
- 99.7% of values are within 3σ of the mean.
How do you find the 80th percentile of a normal distribution?
- Find Z such that P(Z ≤ z) = 0.80 using the Z-table.
- Convert back to X using:
X = μ + Zσ
Example: If ( X ∼ N(100, 225) ), what is the 80th percentile?
- Find Z ≈ 0.84 from the Z-table.
- Use the formula:
X = 100 + (0.84 × 15) = 112.6
What happens when you sum two independent normal variables?
The sum is also normally distributed:
S = X₁ + X₂ ∼ N(μ₁ + μ₂, σ₁² + σ₂²)
What is the formula for the probability density function (PDF) of a normal distribution?
f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}
where: \n μ is the mean (center of the distribution). \n σ^2 is the variance (square of the standard deviation). \n e is Euler’s number (~2.718). \n π is the mathematical constant (~3.1416).
