STAT 101 Lecture Notes - Lecture 12: Central Limit Theorem, Random Variable, Standard DeviationPremium
Course CodeSTAT 101
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Stat 101 - Introduction to Business Statistics - Lecture 12: The Normal Distribution
Facts we already know about normal distributions:
● 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
○ 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.
● 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)
○ Green: original
○ Purple: location shift 3 left
○ Yellow: scale change 2
○ Blue: location shift 6 right and scale
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