ESSENTIAL STATISTICS IN BUSINESS AND
ECONOMICS 3RD EDITION DOANE
TESTED QUESTIONS AND COMPLETE
SOLUTIONS
◉ Expected Value of a Function of Two Random Variables
Answer: If g(Y₁, Y₂,..., Yk) is a function of DRVs with a probability
function p(y₁, y₂, ..., yk),
E[g(y₁, y₂, ..., yk)] = ∑(all yk)...∑(all y2)∑(all y₁) g(y₁, y₂, ...,
yk)p(y₁, y₂, ..., yk).
For functions of CRVs,
E[g(y₁, y₂, ..., yk)] = ∫(-∞ to ∞)...∫(-∞ to ∞)∫(-∞ to ∞) g(y₁, y2, ...,
yk)f(y1, y₂, ..., yk) dy₁ dy₂ ... dyk
◉ Special Theorems for the Expected Value of Two Random
Variables
Answer: E(c) = c
E(c * g(Y₁, Y₂) = c * E[g(Y₁, Y₂)]
E[g1(Y₁, Y₂) + g2(Y₁, Y₂) + ... + gk(Y₁, Y₂)] = E[g1(Y₁, Y₂)] +
E[g₂(Y₁, Y2)] + ... + E[gk(Y₁, Y₂)]
E[g(Y₁)h(Y₂)] = E[g(Y₁)] * E[h(Y2)], if Y₁ and Y₂ are independent
◉ Covariance of Two Random Variables
,Answer: Cov(Y₁, Y₂) = E[Y₁ - µ₁)(Y₂ - µ₂)] = E(Y₁Y₂) - E(Y1)E(Y₂) =
E(Y₁Y₂) - µ1µ2
◉ Covariance of Independent Random Variables
Answer: Cov(Y₁, Y2) = 0
(i.e. independent variables are uncorrelated)
◉ Linear Function of Random Variables
Answer: U1 = a₁Y₁ + a₂Y₂ + ... + anYn = ∑(i=1 to n) aiYi,
where a₁ .. an are coefficients, and Y1 ... Yn are variables on a
sample space
◉ Properties of Linear Functions of Random Variables
Answer: E(U1) = ∑(i=1 to n) a₋i * µ₋i
V(U1) = ∑(i=1 to n) a₋i²*V(Y₋i) + 2∑∑(1 ≤ i < j ≤ n) a₋i * a₋j *
Cov(Y₋i, Y₋j)
Cov(U₋i, U₋j) = ∑(i=1 to n)∑(j=1 to m) a₋i * b₋j * Cov(Y₋i, Y₋j),
where the double sum is over all pairs (i, j) with i < j.
◉ Correlation Coefficient
Answer: If σ₁ and σ₂ are the variances of Y₁ and Y₂, the correlation
coefficient is:
ρ = Cov(Y₁, Y₂) / σ1σ2, -1 ≤ ρ ≤ 1.
,If |ρ| = 1, P[aX + bY + c] = 1
◉ Multinomial Experiment
Answer: An experiment with n independent trials, where each
trial may produce one of k outcomes. We are interested in the
random variables Y₁, Y₂, ..., Yk, where Yi is the number of trials
that produce outcome i.
∑(i=1 to k) p₋i = 1 and ∑(i=1 to k) Y₋i = n.
◉ Multinomial Joint Probability Function
Answer: p(y₁, y₂, ..., yk) = P[Y₁ = y₁, ..., Y₋k = yk] = (n! / y₁!y₂!...yk!)
* p₁^(y₁) *...*p₋k^(y₋k),
where ∑(i=1 to k) p₋i = 1 and ∑(i=1 to k) y₋i = n.
(the binomial distribution is a special case of this where there are
k = 2 outcomes)
◉ Chi-Square Distribution
Answer: A gamma distributed random variable with α = v/2, and
β=2
(v is the # of degrees of freedom)
◉ context
, Answer: ideally tells who was measured, what was measured, how
the data were collected, where the data were collected, and when
and why the study was performed
◉ data
Answer: systematically recorded information, whether numbers
or labels, together with its context
◉ data table
Answer: an arrangement of data in which each row represents a
case and each column represents a variable
◉ case
Answer: an individual about whom or which we have data
◉ variable
Answer: holds information about the same characteristic for many
cases
◉ categorical variable
Answer: a variable that names categories (whether with words or
numerals)
◉ quantitative variable
ECONOMICS 3RD EDITION DOANE
TESTED QUESTIONS AND COMPLETE
SOLUTIONS
◉ Expected Value of a Function of Two Random Variables
Answer: If g(Y₁, Y₂,..., Yk) is a function of DRVs with a probability
function p(y₁, y₂, ..., yk),
E[g(y₁, y₂, ..., yk)] = ∑(all yk)...∑(all y2)∑(all y₁) g(y₁, y₂, ...,
yk)p(y₁, y₂, ..., yk).
For functions of CRVs,
E[g(y₁, y₂, ..., yk)] = ∫(-∞ to ∞)...∫(-∞ to ∞)∫(-∞ to ∞) g(y₁, y2, ...,
yk)f(y1, y₂, ..., yk) dy₁ dy₂ ... dyk
◉ Special Theorems for the Expected Value of Two Random
Variables
Answer: E(c) = c
E(c * g(Y₁, Y₂) = c * E[g(Y₁, Y₂)]
E[g1(Y₁, Y₂) + g2(Y₁, Y₂) + ... + gk(Y₁, Y₂)] = E[g1(Y₁, Y₂)] +
E[g₂(Y₁, Y2)] + ... + E[gk(Y₁, Y₂)]
E[g(Y₁)h(Y₂)] = E[g(Y₁)] * E[h(Y2)], if Y₁ and Y₂ are independent
◉ Covariance of Two Random Variables
,Answer: Cov(Y₁, Y₂) = E[Y₁ - µ₁)(Y₂ - µ₂)] = E(Y₁Y₂) - E(Y1)E(Y₂) =
E(Y₁Y₂) - µ1µ2
◉ Covariance of Independent Random Variables
Answer: Cov(Y₁, Y2) = 0
(i.e. independent variables are uncorrelated)
◉ Linear Function of Random Variables
Answer: U1 = a₁Y₁ + a₂Y₂ + ... + anYn = ∑(i=1 to n) aiYi,
where a₁ .. an are coefficients, and Y1 ... Yn are variables on a
sample space
◉ Properties of Linear Functions of Random Variables
Answer: E(U1) = ∑(i=1 to n) a₋i * µ₋i
V(U1) = ∑(i=1 to n) a₋i²*V(Y₋i) + 2∑∑(1 ≤ i < j ≤ n) a₋i * a₋j *
Cov(Y₋i, Y₋j)
Cov(U₋i, U₋j) = ∑(i=1 to n)∑(j=1 to m) a₋i * b₋j * Cov(Y₋i, Y₋j),
where the double sum is over all pairs (i, j) with i < j.
◉ Correlation Coefficient
Answer: If σ₁ and σ₂ are the variances of Y₁ and Y₂, the correlation
coefficient is:
ρ = Cov(Y₁, Y₂) / σ1σ2, -1 ≤ ρ ≤ 1.
,If |ρ| = 1, P[aX + bY + c] = 1
◉ Multinomial Experiment
Answer: An experiment with n independent trials, where each
trial may produce one of k outcomes. We are interested in the
random variables Y₁, Y₂, ..., Yk, where Yi is the number of trials
that produce outcome i.
∑(i=1 to k) p₋i = 1 and ∑(i=1 to k) Y₋i = n.
◉ Multinomial Joint Probability Function
Answer: p(y₁, y₂, ..., yk) = P[Y₁ = y₁, ..., Y₋k = yk] = (n! / y₁!y₂!...yk!)
* p₁^(y₁) *...*p₋k^(y₋k),
where ∑(i=1 to k) p₋i = 1 and ∑(i=1 to k) y₋i = n.
(the binomial distribution is a special case of this where there are
k = 2 outcomes)
◉ Chi-Square Distribution
Answer: A gamma distributed random variable with α = v/2, and
β=2
(v is the # of degrees of freedom)
◉ context
, Answer: ideally tells who was measured, what was measured, how
the data were collected, where the data were collected, and when
and why the study was performed
◉ data
Answer: systematically recorded information, whether numbers
or labels, together with its context
◉ data table
Answer: an arrangement of data in which each row represents a
case and each column represents a variable
◉ case
Answer: an individual about whom or which we have data
◉ variable
Answer: holds information about the same characteristic for many
cases
◉ categorical variable
Answer: a variable that names categories (whether with words or
numerals)
◉ quantitative variable
