P&s 2

Front 1

Random Variable

Back 1

Random Variable X is any* function X: S --» R. (from sample space to rational numbers)

*X must be measurable ({X<t} must belong to F)

Front 2

Discrete r.v.

Back 2

A r.v. is discrete if it assumes only finite or infinitely countable amount of values. x1, x2, ...

Front 3

Probability Mass Function

Back 3

p.m.f.

p_x(a) = P(X=a), aєR

Front 4

Cumulative Function

Back 4

c.d.f.

F_x(t) = P(X<=t)

Front 5

Properties of cdf

Back 5

0 <= F_x(t) <= 1

F_x(t) is non-decreasing

· lim(F_x(t)) = 0 (for t --» -inf)

· lim(F_x(t)) = 1 (for t --» +inf)

F_x(t) = F_x(t+0) = F_x(t+e), e --»0

but

F _x(t) ≠ F_x(t-0)

Hint 5

Bounds

Non-decreasing (stairs up)

Right-continuity

Front 6

pdf vs cdf

Back 6

Hint 6

cdf as sum of pdf-s

pdf as cdf(t) - cdf(t-0)

Front 7

Bernoulli r.v.

Back 7

X=0 or X=1 (indicator of success)

p_x(a) = P(X=a) = p (if X=1)

1-p (if X=0)

F_x(a) :

1-p

1

Front 8

Binomial r.v.

Back 8

Bernoulli repeated n times

Binomial ~ num of succ (1)

p_x(a) = P(X=a) = C(a, n) · p^{a} · (1-p)^{n-a}

sum p_x(a) = 1 ( by binomial theorem)

Front 9

Geometric r.v

Back 9

Numbers of Bernoulli experiments before success

p_x(a) = p · (1-p)^{a-1}

Front 10

Poisson r.v

Back 10

limit of binomial as n --» inf

assume parameter lambda = n·p

p_x(a) = P(X=a) = lambda^{a} · e^{-lambda} / a!

Hint 10

when event happens very rarely but number of experiments is very big

Front 11

Conditional distribution

Back 11

p_{X|Y} (x|y) = P{X=x|Y=y} = P{X=x, Y=y} / P{Y=y}

for a fixed value y is a pmf of

r.v {X|Y=y} which has conditional distribution

Front 12

Expected value

Back 12

Assume X is a r.v taking on values x1,x2,... . If sum{x_i · p(x_i)} < inf, then expected value E(X) = sum{x_i · p(x_i)}

Front 13

Expected values for different distributions

Back 13

Binomial

Bernoulli

Geometric

Poisson

Exponential

Normal

Front 14

Independent r.v-s

Back 14

Random Variables X and Y are independent if for every values (x, y) evens A:={X=x} and B:={Y=y} are independent

Front 15

Independence by p.m.f

Back 15

p_{X,Y} (x, y) = p_X (x) · p_Y (y)

Front 16

Expectation of g(X)

Back 16

E{Y} = E{g(X)} = sum {g(x_i) · p(x_i)}

Front 17

Linearity of expectation

Back 17

E{a·X + b·Y} = a·E{X} + b·E{Y}

Front 18

Total probability rule for expectations

Back 18

If E_1...E_n - partition of the sample space, then:

E{X} = E{X|E_1}·P{E_1} + ... + E{X|E_n}·P{E_n}

Front 19

k_th moment of X

Back 19

If sum{|x|^k · p(x) } < inf

mu_k = E{X^k}

Front 20

k-th central moment

Back 20

If sum{|x|^k · p(x)} < inf

K-th Central moment = E( (X-E(X))^k )

Front 21

k-th factorial moment of X

Back 21

If sum {|x|^k · p(x) } < inf:

mu^(k) = E( X·(X-1)·(X-2)·...·(X-k+1) )

Front 22

Variance

Back 22

sigma_2 := Var(X) = E( ( X - E(X) )^2 )

Hint 22

measure of disperse/spread around tge mean

Front 23

Standard deviation

Back 23

square root of Var(X)

Front 24

Properties of variance

Back 24

E( (X - E(X))^2 ) = E(X^2) - E(X)^2

Var(a·X) = a^2 · Var(X)

Front 25

Variances for different distributions

Back 25

Binomial -- np(1-p)

Front 26

Expectation of the product

Back 26

If X abd Y are independent:

E(X·Y) = E(X) · E(Y)

Front 27

Covariance

Back 27

Cov(X,Y) = E( (X - E(X)) · (Y - E(Y)) )=

= E(X·Y) - E(X)·E(Y)

If Cov(X,Y) > 0:

X and Y are dependant

Positive correlation:

larger values of X usually respond to larger values of Y

Front 28

Properties of Covariance

Back 28

If X and Y are independent:

Cov(X,Y) = 0

Var(X+Y) = Var(X) + 2·Cov(X,Y) + Var(Y)

Front 29

Variance of the sum

Back 29

Only if X and Y are independent:

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

Front 30

Correlation coefficient

Back 30

Correlation coefficient between X and Y = Cor(X,Y) = Var(X,Y) / sigma(X)·sigma(Y)

Front 31

Properties of correlation

Back 31

|Cor(X, Y)| <= 1

Cor(X, Y) = ±1 --» Y = aX + b

Hint 31

Cauchy-Bunyakivsky-Shwartz inequality)