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Lecture 12

# STAT 101 Lecture Notes - Lecture 12: Central Limit Theorem, Random Variable, Standard DeviationPremium

Department
Statistics
Course Code
STAT 101
Professor
Richard Waterman
Lecture
12

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Stat 101 - Introduction to Business Statistics - Lecture 12: The Normal Distribution
The shape of the distribution (bell curve)
It is characterized by its mean, µ, and variance, σ2
if you know this, you can create the entire distribution
How to calculate certain probabilities, as prescribed by the Empirical Rule.
We have seen that both Binomial and Poisson distributions start to look normally
distributed:
For the Binomial, when n gets large.
For the Poisson, when λ, the rate, gets large.
This convergence to normality, can be explained with the Central Limit Theorem.
Height
Probabilities for a normal random variable are calculated by finding the area under this
curve. That is, integration is used.
Because we don’t use integration, we can instead use pre-calculated values, aka the z-
table (or calculator)
Shifts & Scale Changes
The mean µ controls the location of the center of the distribution.
The mean, µ can take on any value between −∞ and +∞.
The variance, σ 2 controls how spread out the distribution is.
The greater the variance, the greater the spread.
σ is always greater than or equal to 0.
A key feature of the normal is that in general, there is no relationship between µ and σ.
Contrast this fact to the binomial and poisson, where the variance is linked to the mean.
NOTE: in steins book and in this class, the second number that appears when
describing distributions is variance but in other books/classes it can be sigma
ie: in this class (mean, variance)
● Changes:
Green: original
Purple: location shift 3 left
Yellow: scale change 2
Blue: location shift 6 right and scale
change ½