Stats Review

Question 1

Random Variable

Answer 1

Variable whose values we do not know with certainty

Question 2

Probability

Answer 2

A function defined over the possible values a random variable can take on.

Between 0 and 1

Add up to one

Question 3

Discrete Random Variable

Answer 3

Finite or countably infinite

Question 4

probability density function (pdf) Discrete

Answer 4

Summarizes the information concerning the possible outcomes of the random variable and the corresponding probabilities:

f (xj) = pj, j = 1, …, k.

Question 5

continuous random variable

Answer 5

Any real value with zero probability, so many possible values that we cannot count them or match them
up with the positive integers

Question 5

PDF Continuous

Answer 5

P(a ≤ X ≤ b) = ∫fX(x)dx

continuous random variable X as fX (x)

Question 6

cumulative distribution function

Answer 6

F(x) ≡P(X ≤ x)

• For discrete variables, the cdf is obtained by summing the pdf over all values xj such that
  xj ≤ x. For a continuous random variable, the cdf is the area under the pdf, fX, to the left of
  the point x, that is F(x) = ∫ x−∞ fX (x)dx.
• Because F(x) is a probability, it is always between 0 and 1
• For x1 ≤ x2, F(x1) ≤ F(x2), that is F(x) is an increasing function of x
• For any number c, P(X > c) = 1 − F(c)
• For any numbers a < b, P(a < X ≤ b) = F(b) − F(a)

Question 7

Expected Value

Answer 7

a measure of central ten-
dency of its probability distribution.

a weighted average
of all possible values of X, where the weights are determined by the pdf

The expected value is also called population mean.

The expected value can sometimes be denoted by μX or μ.

For any constant c, E(cX) = cE(X).
For any constants a and b, E(aX + b) = aE(X) + b.

If {a1, a2, …, an} are constants and {X1, X2, …, Xn} are random variables, then
E(a1 X1 + a2 X2 + … + an Xn) =a1 E(X1) + a2 E(X2) + … + an E(Xn).

Question 8

E(X) Finite

Answer 8

E(X) =x1 f (x1) + x2 f (x2) + … +xk f (xk) ≡ ∑xj f (xj)

Question 9

E(X) continuous

Answer 9

If X is a continuous random variable and fX (x) is its pdf, then:
E(X) = ∫xf (x)dx

Question 10

Variance

Answer 10

How far X is from its mean μ, on average

Var(X) ≡E[(X − μ)^2]

Denoted by σ^2

Question 11

Standard Deviation

Answer 11

σ, sqrt(var)

Question 12

standardized random variable

Answer 12

Z ≡ (X − μ)/σ

Question 13

Joint Distribution

Answer 13

F(x, y) ≡P(X ≤ x, Y ≤ y)

Question 14

X and Y are said to be independent iff

Answer 14

fX,Y (x, y) = fX (x) fY (y)

Question 15

Conditional Distribution

Answer 15

The joint distribution tells us the probability that certain values of two (or more) random variables are observed concurrently.

fY|X (y|x) = (fX,Y (x, y))/(fX (x))

Question 16

Covariance

Answer 16

The covariance measures the extent of the linear association between two random variables. If the
covariance is positive (negative), the two variables move in the same (opposite) directions.

Cov(X, Y) ≡E[(X − μX )(Y − μY )]

σXY

If X and Y are independent Cov(X,Y) = 0, but if Cov(X,Y) = 0 that does not imply that X and Y are independent

Question 17

Correlation Coefficient

Answer 17

Corr(X,Y) = σ^2XY/σX σY

Question 18

Sums of random variables

Answer 18

Var(aX + bY) =a^2Var(X) + b^2Var(Y) + 2abCov(X, Y)

Question 19

Conditional Expectations

Answer 19

(Y|x) =∑yj fY|X (yj|x) (Discrete)

E(Y|x) = ∫y fY|X (y|x)dy (Continuous)

Question 20

law of iterated expectations

Answer 20

E[E(Y|X)] = E(Y)

Question 21

Estimator

Answer 21

A function of the sample outcomes that generates a value for θ

Question 22

Ybar Estimate

Answer 22

1/n ∑Yi

Question 23

S^2

Answer 23

S^2 = 1/n − 1∑(Yi − Y)^2

Question 24

SXY

Answer 24

1/n − 1∑(Xi − X)(Yi − Y)

Question 25

T-Test

Answer 25

T =√n(Ybar − μ0)/S