Probability

Disjoint events, probability one of them occured is P(1) + P(2)+ ..., it's the same as probability one of number in die is add up to 1.
events : set of outcomes.
Law of Addition Rule, disjoint vs not disjoint. Disjoint, P(A and B) automatically zero.

P (A or B) = P (A) + P (B) − P (A and B)

Probability distributions : all disjoint, between 0 and 1, add up to 1.
Sample space, list of all possible outcomes. die event {2,3}, then complement {1,4,5,6}
Independence can allow two events to occurs at the same time. This will have a multiplication rule, where P( A and B occurs at the same time)

P(A and B) = P(A) × P(B)

For example, Since gender,handedness and person order(first vs third person) are independence event. We can, multiply together. (First or second or third) person female and right hand = P(female) * P(right hand).First two MR, third FL,

P(MR) ** 2  * P(FL)

Marginal probability, row/columns total in propotion of total population. Not consider other variables.
Join probability, take intersection of two variables, in proportion of total population.
*Conditional probability:

P (A|B) = P (A and B) / P(B) **or** P(A and B) = P(A|B) × P(B)

We can use General multiplication rule in conjunction with marginal and conditional probability.
with replacement, data can picked over and over again, thus make it independent every pick. Without replacement make it dependant events.

Distributions

For z-score



In [3]:

    
%%R

mean = 1500
sd = 300
d = 1800
(d-mean)/sd

To calculate the percentiles



In [9]:

    
%%R

mean = 1500
sd = 300
point = 2100
LT = T

pnorm(point,mean=mean,sd=sd,lower.tail=LT)









    





[1] 0.9772499



In [11]:

    
%%R

mean = 1500
sd = 300
percentile = 0.4
LT = T

qnorm(percentile,mean=mean,sd=sd,lower.tail=LT)









    





[1] 1423.996

Pnorm ranges



In [30]:

    
%%R

mean = 70
sd = 3.3
lower = 69
upper = 74
LT = T
# IN RANGE
# 1 - pnorm(upper,mean=mean,sd=sd,lower.tail=!LT) - pnorm(lower,mean=mean,sd=sd,lower.tail=LT)

# OUT RANGE
# pnorm(upper,mean=mean,sd=sd,lower.tail=!LT) + pnorm(lower,mean=mean,sd=sd,lower.tail=LT)









    





[1] 0.5063336

Standard normal distributions, is mean 0 and sd 1. Used in default pnorm and qnorm R.
Always draw normal probability distributions, and shade the are we're interesed.
In evaluation the distribution through normal probability plot, imagine a straight line. If the data is skewed to the high end of the line, it's right skewed. Otherwise , low-end is left skewed.
To construct normal probability plot, for each observation, make a scatter plot with its Z-score(horizontal) against its percentile(vertical).



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