The University of Texas at Austin

ECO 329 (Economic Statistics)

MIDTERM #2 “CHEAT SHEET”

Discrete random variables: univariate

Random variable: maps each outcome of the sample space S to a number

Discrete random variable: random variable with finite or countable set of possible values {x∗k }
Probability mass function (pmf): pX (x∗k ) = P (X = x∗k ) for every possible outcome x∗k
Cumulative distribution function (cdf): FX (x0 ) = P (X ≤ x0 ) = pX (x∗k )
P
x∗k ≤x0

Population mean or population average or expected value: µX = E(X) = x∗k pX (x∗k )

P
k
Population variance: σX
2 = V ar(X) = E[(X − µ )2 ] =
k (xk
∗ − µX )2 pX (x∗k )
P
X
q qP
Population standard deviation: σX = 2 =
σX k (xk
∗ − µX )2 pX (x∗k )
Linear transformation of a random variable: Y = a + bX

• Population mean: µY = a + bµX

• Population variance: σY2 = b2 σX

2

• Population standard deviation: σY = |b|σX

X−µX
Standardized random variable: Z = σx has µZ = 0 and σZ2 = σZ = 1
Bernoulli random variable: X ∼ Bernoulli(π), where π = P (X = 1) and 1 − π = P (X = 0)

• µX = π, σX
2 = π(1 − π), σ =
p
X π(1 − π)

Binomial random variable: X ∼ Binomial(n, π), where X is the total number of 1’s (“successes”) from n
independent trials of the Bernoulli(π) random variable

• pmf: P (X = x) = n
x
x π (1 − π)n−x for x ∈ {0, 1, 2, . . . , n}

• µX = nπ, σX
2 = nπ(1 − π), σ =
p
X nπ(1 − π)

“Sample proportion” random variable: P = n1 X, where X ∼ Binomial(n, π)

q
π(1−π) π(1−π)
• µP = π, 2
σX = n , σX = n

Discrete random variables: bivariate

Joint probability function: pXY (x, y) = P (X = x ∩ Y = y) = P (X = x, Y = y)

• pXY (x∗k , y`∗ ) ≥ 0 for any possible outcome pair (x∗k , y`∗ )

• pXY (x∗k , y`∗ ) = pXY (x∗k , y`∗ ) = 1

P PP
(k,`) k `

Joint cumulative distribution function: FXY (x, y) = P (X ≤ x ∩ Y ≤ y) = P (X ≤ x, Y ≤ y)

1
Obtaining marginal probability functions from the joint probability distribution:

• pX (x∗k ) = ` pXY (xk , y` )

∗ ∗ and pY (y`∗ ) = pXY (x∗k , y`∗ )
P P
k

pXY (x,y)
Conditional probability function (of X given Y ): pX|Y (x|y) = P (X = x|Y = y) = pY (y)
Conditional mean (of X given Y ): µX|Y =y`∗ = E(X|Y = y`∗ ) = x∗k pX|Y (x∗k |y`∗ )
P
k

Conditional variance (of X given Y ): σX|Y

2
=y ∗ = V ar(X|Y = y` ) =
∗
k (xk
∗ − µX|Y =y`∗ )2 pX|Y (x∗k |y`∗ )
P
`

Population covariance: σXY = Cov(X, Y ) = (k,`) (xk − µX )(y` − µY ) pXY (xk , y` )

∗ ∗ ∗ ∗
P

Population correlation: ρXY = Corr(X, Y ) = σXY

σX σY (−1 ≤ ρXY ≤ 1; sign(ρXY )=sign(σXY ))
Independence:

• X and Y are independent if and only if pXY (x∗k , y`∗ ) = pX (x∗k )pY (y`∗ ) for every possible outcome pair
(x∗k , y`∗ ). If this equality fails for any (x∗k , y`∗ ) pair, then X and Y are dependent.

• X and Y are independent if and only if:

– For any possible value y`∗ , the conditional distribution of X given Y = y`∗ is the same as the
unconditional (marginal) distribution of X. (And the same when switching roles of X and Y .)

Linear transformations of random variables: V = a + bX, W = c + dY

• Covariance: σV W = bdσXY

• Correlation: ρV W = ρXY if bd ≥ 0 and −ρXY if bd < 0

Linear combination of two random variables: V = k + aX + bY

• Mean: µV = k + aµX + bµY

• Variance: σV2 = a2 σX
2 + b2 σ 2 + 2abσ
Y XY

– Special case (a = b = 1; V = X + Y ): σV2 = σX

2 + σ 2 + 2σ
Y XY
– Special case (a = 1, b = −1; V = X − Y ): σV2 = σX
2 + σ 2 − 2σ
Y XY
q q
• Standard deviation: σV = σV2 = 2 + b2 σ 2 + 2abσ
a2 σX Y XY

Continuous random variables: univariate

Continuous random variable: random variable taking on values within an interval of the real line, multiple
intervals of the real line, or the entire real line (with an uncountable set of possible outcomes)
´b
Probability density function (pdf): function fX (x) such that P (a ≤ X ≤ b) = a fX (x) dx
´ x0
Cumulative distribution function (cdf): FX (x0 ) = P (X ≤ x0 ) = −∞ fX (x) dx
Relationship between pdf and cdf: fX (x0 ) = FX0 (x0 )
Population quantile τX,q : for any q between 0 and 1, τX,q is value such that P (X ≤ τX,q ) = FX (τX,q ) = q
´∞
Population mean or population average or expected value: µX = E(X) = −∞ xfX (x) dx
´
2 = V ar(X) = E[(X − µ )2 ] = ∞ (x − µ )2 f (x) dx
Population variance: σX X −∞ X X
q q´
∞
Population standard deviation: σX = sd(X) = σX 2 =
−∞ (x − µX ) fX (x) dx
2

2
Symmetry: X has a symmetric distribution if, for a midpoint x∗ , fX (x∗ − v) = fX (x∗ + v) for all v ≥ 0.
Linear transformations and combinations: same formulas hold as in the discrete X case
Uniform distribution with lower bound a and upper bound b: fX (x) = 1
b−a for a ≤ x ≤ b and 0 otherwise

Normal random variables

√1 e− 2 ( σ )
1 x−µ 2
Normal distribution: X ∼ N (µ, σ 2 ) has pdf fX (x) = σ 2π
for −∞ < x < ∞ with E(X) =
µ, V ar(X) = σ 2
Standard normal: normal with µ = 0 and σ 2 = σ = 1, that is Z ∼ N (0, 1)
X−µ
Standardized normal RV: X ∼ N (µ, σ 2 ) implies σ ∼ N (0, 1) = Z and X = µ + σZ
Probability intervals for the standard normal (Z ∼ N (0, 1)):

• P (−1.96 < Z < 1.96) ≈ 0.95 (that is, z0.025 = 1.96)

• P (−1.645 < Z < 1.645) ≈ 0.90 (that is, z0.05 = 1.645)

Linear combination of normal random variables is a normal random variable (figure out mean and variance
from the linear combination formulas)
Log-normal random variable: X is log-normal if ln(X) ∼ N (µ, σ 2 ) for some µ and σ 2

Continuous random variables: bivariate

´d´b
Joint pdf of X and Y : function fXY (x, y) such that P (a ≤ X ≤ b, c ≤ Y ≤ d) = c a fXY (x, y) dx dy
Population covariance: σXY = Cov(X, Y ) = E[(X − µX )(Y − µY )] (don’t worry about integral form)
Population correlation: ρXY = Corr(X, Y ) = σXY
σX σY (−1 ≤ ρXY ≤ 1; sign(ρXY )=sign(σXY ))
Independence of continuous RV’s: X and Y independent if and only if fXY (x, y) = fX (x)fY (y)

• Independence implies σXY = ρXY = 0 (but not vice versa)

• Independence implies P (a ≤ X ≤ b, c ≤ Y ≤ d) = P (a ≤ X ≤ b) · P (c ≤ Y ≤ d)

Sampling distributions

i.i.d. random variables: X1 , X2 , . . . , Xn are independent and identically distribution (i.i.d.) if they are
mutually independent and have the same distribution function (FXi (x) = FX (x) for all i ∈ {1, 2, . . . , n}
and all x)

Sampling distribution of the sample average (X̄ = n1 (X1 + X2 + . . . + Xn ) for i.i.d. RV’s)

• Xi ∼ Bernoulli(π) implies X̄ ∼ n1 Binomial(n, π)

 2

• Xi ∼ N (µ, σ 2 ) implies X̄ ∼ N µ, σn

s2
Sampling distribution of the sample variance for i.i.d. normal random variables: (n − 1) σX2 ∼ χ2n−1 , where
X
χ2n−1 has a chi-squared distribution with n − 1 degrees of freedom. (If Z1 , . . . , Zk are i.i.d. N (0, 1), then it
is said that Z12 + · · · + Zk2 has a χ2k distribution.)