Class Notes (1,100,000)

US (480,000)

UPenn (2,000)

STAT (70)

STAT 101 (30)

Richard Waterman (10)

Lecture 12

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

Department

StatisticsCourse Code

STAT 101Professor

Richard WatermanLecture

12This

**preview**shows half of the first page. to view the full**3 pages of the document.**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

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

###### You're Reading a Preview

Unlock to view full version

Subscribers Only