definition or RV and expected and variance (week 3)

Question 1

What is Random Variable?

Answer 1

rule or function that assigns a real number to every outcome of an experiment

Question 2

The Probability Mass Function (PMF)

Answer 2

formula of the probability X, rather than listing all of them one by one

Question 3

The Cumulative Distribution Function (CDF) properties (3)

Answer 3

1. F is non decreasing
2. F(-infinity) = 0 and F(+infinity) = 1
3. F may have countable number of jump discontinuities and its right

Question 4

PMF graph looks like

Answer 4

straight line

Question 5

CDF graph looks like

Answer 5

stairs

Question 6

joint distribution

Answer 6

PMF = Px,y (x,y) = P((X=x) n (Y=y))
CDF = Fx,y (x,y) = P((X<= x) n (Y<= y))

Question 7

Non-linear transformation of a discrete random variable

Answer 7

example:
X has 3 possible values -1,0,1 with probabilites 0.2, 0.3,0.5 respectively, g(X) = X^2. g(X) has only 2 possible values 0 and 1 with probabilites 0.3 and 0.7

This is bcs g is a function from R -> R that is well - defined at all points in the support of X, then g(X) will hv possible values g(x1) to g(xk) with probabilites p1 to pk.

jdi kan -1^ 2 = 1 nah prob 1 itu ada 0.5 and prob -1 itu 0.2 jdi mrka ditambah

Question 8

Variance formula of RV

Answer 8

E(X-mean x)^2

Question 9

Standard Deviation formula

Answer 9

sqrt(variance)

Question 10

Volatility

Answer 10

Standard deviation of the return to a risky asset

Question 11

var vs sd?

Answer 11

sd use the same unit measurement, so easier to communicate

Question 12

Var if have constant

Answer 12

Var(a + bX) = b^2 Var(X)