Discrete Probability Distributions, Statistical
Inferencing Standard Normal Distribution Z =
x μ
σ
Discrete: Countable number of values. A, B, Number of away from
Continuous: Uncountable infinite. 1, 1.5, Z Distribution: =1, =0
Simple X P(X) Xi X*P(Xi) Normal approximation to binomial distribution.
Event =n , =√n (1- )
AA 2 0.25 x1 0.5 P(X=x) ≅ P(x-0.5<Y<x+0.5)
AB 1 0.25 X2 0.25 Convert to Z and solve.
BA 1 0.25 X3 0.25
BB 0 0.25 X4 0
Sampling Distributions
Xi*P(Xi) = E(X) = 1
P(X=1) = P(AB)+P(BA) = 0.25+0.25 Sample from population.
P(X≥1) = P(AA)+P(AB)+P(BA) = 0.5+0.25+0.25 of population is unknown. x̄ = Sample mean. Random
Variable
σ
n
σ x̄ =
E(X) = = ∑ xi p(xi ) √n Standard error.
i=1
Binomial Distribution: Two outcomes Central Limit Theorem.
E(X) = n ( = Proportion) When n>30 assume sampling distribution is approximately
Type I, error. Reject H0 when true. Normal Distribution.
Type II, error. Do not reject when false.
P-value: Probability of type I error.
Z=
x̄ μ
σ x̄
Variance 2 = ∑ (xi μ) p(X)
i=1
2
n
Estimation
Bivariate Distribution
Join probability distribution of two variables. X̄ to Point Estimate
X\Y 1 2 3 4 Total Y ± number of standard errors x̄ to give Confidence Interval:
1 P(1,1) P(1,2) P(1,3) P(1,4) 0.25 LCL/ULC
2 P(2,1) P(2,2) P(2,3) P(2,4) 0.5 x̄±z* x̄
3 P(3,1) P(3,2) P(3,3) P(3,4) 0.125 1- = CI for x̄
( -z* x̄)<x̄<( +z* x̄)
4 P(4,1) P(4,2) P(4,3) P(4,4) 0.125
Rearrange for CI of
Total 0.125 0.5 0.25 0.125 Must be
(x̄-z* x̄)< <(x̄+z* x̄)
X 1
If is unknown, use t-distribution
Marginal probabilities.
s ≅ Sample Standard Deviation
Covariance measure of the direction/slope.
s
sx̄ =
√n
Sx̄ Sample Standard Error
COV(X,Y) = xy = ∑ ∑(x μ )(y μ )P (x, y) =
x y
tdf ,n1 =
x̄ μ
sx̄
∑ ∑ xyp(x, y )
all x all y
μ μx y
1- = CI for
x̄±tdf,a/2*sx̄
Coefficient of correlation measures the strength of Minimum sample size (n) for margin of error (B)
relationship.
n=( *
z a/2 σ 2
B )
σ xy
= r = σx σy
Continuous Probability Distributions Hypothesis Testing (I)
Normal Distribution H0: Null Hypothesis, no change.
Even, symmetrical distribution around the mean. Area under H1: Alternative Hypothesis. Opposite of H0, conclusion if H0
the curve = 1. -∞<x<∞ is rejected.
Larger = Flatter curve, larger dispersion. 6 steps
Inferencing Standard Normal Distribution Z =
x μ
σ
Discrete: Countable number of values. A, B, Number of away from
Continuous: Uncountable infinite. 1, 1.5, Z Distribution: =1, =0
Simple X P(X) Xi X*P(Xi) Normal approximation to binomial distribution.
Event =n , =√n (1- )
AA 2 0.25 x1 0.5 P(X=x) ≅ P(x-0.5<Y<x+0.5)
AB 1 0.25 X2 0.25 Convert to Z and solve.
BA 1 0.25 X3 0.25
BB 0 0.25 X4 0
Sampling Distributions
Xi*P(Xi) = E(X) = 1
P(X=1) = P(AB)+P(BA) = 0.25+0.25 Sample from population.
P(X≥1) = P(AA)+P(AB)+P(BA) = 0.5+0.25+0.25 of population is unknown. x̄ = Sample mean. Random
Variable
σ
n
σ x̄ =
E(X) = = ∑ xi p(xi ) √n Standard error.
i=1
Binomial Distribution: Two outcomes Central Limit Theorem.
E(X) = n ( = Proportion) When n>30 assume sampling distribution is approximately
Type I, error. Reject H0 when true. Normal Distribution.
Type II, error. Do not reject when false.
P-value: Probability of type I error.
Z=
x̄ μ
σ x̄
Variance 2 = ∑ (xi μ) p(X)
i=1
2
n
Estimation
Bivariate Distribution
Join probability distribution of two variables. X̄ to Point Estimate
X\Y 1 2 3 4 Total Y ± number of standard errors x̄ to give Confidence Interval:
1 P(1,1) P(1,2) P(1,3) P(1,4) 0.25 LCL/ULC
2 P(2,1) P(2,2) P(2,3) P(2,4) 0.5 x̄±z* x̄
3 P(3,1) P(3,2) P(3,3) P(3,4) 0.125 1- = CI for x̄
( -z* x̄)<x̄<( +z* x̄)
4 P(4,1) P(4,2) P(4,3) P(4,4) 0.125
Rearrange for CI of
Total 0.125 0.5 0.25 0.125 Must be
(x̄-z* x̄)< <(x̄+z* x̄)
X 1
If is unknown, use t-distribution
Marginal probabilities.
s ≅ Sample Standard Deviation
Covariance measure of the direction/slope.
s
sx̄ =
√n
Sx̄ Sample Standard Error
COV(X,Y) = xy = ∑ ∑(x μ )(y μ )P (x, y) =
x y
tdf ,n1 =
x̄ μ
sx̄
∑ ∑ xyp(x, y )
all x all y
μ μx y
1- = CI for
x̄±tdf,a/2*sx̄
Coefficient of correlation measures the strength of Minimum sample size (n) for margin of error (B)
relationship.
n=( *
z a/2 σ 2
B )
σ xy
= r = σx σy
Continuous Probability Distributions Hypothesis Testing (I)
Normal Distribution H0: Null Hypothesis, no change.
Even, symmetrical distribution around the mean. Area under H1: Alternative Hypothesis. Opposite of H0, conclusion if H0
the curve = 1. -∞<x<∞ is rejected.
Larger = Flatter curve, larger dispersion. 6 steps
