Midterm Flashcards
Probability Density Function
Expected Value
Expected Value: E = ∑xf(x)
Probability Density Function
Variance
σ² = E(x²) - (E(x))²
Linear Transformation
Linear transformation
Y = a + bX
Linear Transformation
Expected Value
μy = a + bμx
Linear Transformation
Variance
σ² = b²σ²x
Normal Distribution of X
Distribution
x ~ N(μ,σ²)
Normal Distribution of X (population)
Z = ?
Z = (X - μ)/σ ~N(0,1)
Sample normal distribution of X (sample)
Distribution
X̂ ~ N(μ,σ²/n)
Sample normal distribution of X (sample)
Z
z = (X - μ)/ (σ/√n) ~ N(0,1)
Binomial Probability
Probability of k successes
P(Y=k) = (n k)p^k(1-p)^n-k
Binomial Probability
E(Y) =
E(Y) = np
Binomial probability
V(Y) =
V(Y) = np(1-p)
Binomial Probability
Distribution
X~Bin(n,p)
Joint pdf
Covariance X,Y
σx,y = E(XY) - E(X)*E(Y)
Joint pdf
E(XY)
E(XY) = ∑xyh(x,y)
Joint pdf
correlation x,y
ρ = σx,y/σx*σy
Linear combination W
W =
W = a + bX + cY
Linear combination W
Expected value
E(w) = μw = a + bμx + cμy
Linear combination W
Variance
v(x) = σ²w = b²σ²x +2bcσxy + c²σ²y
Test procedure (σ known)
Z test
Z = (X - μ)/ (σ/√n)
Test procedure (s known)
T test
T = (X - μ)/ (S/√n) ~t(n-1)
Test procedure (s known)
CI T test
CI = X ∓ tα/2;n-1 * S/√n
Proportion test
Z = P^ - p / √p(1-p)/n
proportion test
CI
ci =p̂+- zα/2 * √p̂(1-p̂)/n
Variance test
W = (n-1)S²/σ²
Variance test
CI
L = (n-1)S²/xα/2;n-1 U = (n-1)S²/1-xα/2;n-1
Simple linear regression
Basic assumption
E(y|x) = β0 + β1X
Simple linear regression
sample equation
ŷ = B0 + B1X
Simple linear regression
Covariance
Sx,y = 1/n-1 ( ∑XiYi - nXYhat) Also = rxy * Sx * Sy
Simple linear regression
Variance
Sx² = 1/n-1 (∑x²i) - n(x̄)²
Simple linear regression
Slope
B1 = Sx,y/S²x = (rxySxSy)/ S²x
Simple linear regression
Intercept
B0 = Yhat - B1x̄
Simple linear regression
Residual
ei = Yi - Y^i
Simple Linear Regression
SSR
SSR = B1²(n-1)S²x
Simple Linear Regression
correlation
rx,y = Sx,y/Sx*Sy
Simple linear regression
SST =
SST = SSR + SSE = (n-1)Sy^2
Simple linear regression
MSE
MSE = SSE/n-2 = S²e
Simple linear regression
standard error slope
SB1 = Se/√(n-1)*S²x
where Se = √MSE
Simple linear regression
Regression test for slope
T = B1- β1/ SB1 ~ t(n-2)
Simple linear regression
Regression slope CI
CI = B1 +- tα/2;n-2*SB1
Simple linear regression
Coefficient of determination
R² = SSR/SST = 1 - SSE/SST
R²adj = 1 - SSE/SST *n-1/n-k-1
Uniform probability
Distribution
X ~ U(α,β)
Uniform probability
E(X)
E(X) = a+b/2
Uniform probability
V(x)
V(X) = (b-a)²/12
Uniform probability
Probability P(X<=x)
P(X<=x) = x-a/range
Confidence interval standard formula
CI = sample stat +- critical value α/2 * Standard error
MSR =
MSR = SSR / df
Type I error
Wrongfully reject H0
Type II error
Wrongfully accept H0
If the question is about means and you know the POPULATION standard deviation ?
Z test
If the question is about means and you know the SAMPLE standard deviation ?
T test
If the questions is about regression, use the test for the slope.
T = b1- beta1/ SB1 ~ t(n-2)
Proportion test
P^ =
P^ = x/n
Expectation of sample x
E(x̄) =
E(x̄) = μ
Variance of sample x
V(x̄) =
V(x̄t) = σ²/n
Standard deviation of sample x
SD(x̄) =
SD(x̄) = σ/√n
