chapter4 distibution and density functions

Question 1

What are the properties of F(y) and what is F(y) called

Answer 1

1. limit as y goes toe -inf is zero.
   2.limit as y goes to inf is 1
2. if 2 values of y where y1 < y2 then F(y1) <= F(y2)
   F(y) called cumilative distribution function

Question 2

What is f(y)

Answer 2

Probability density function and is the first derivitive of the cumalative distribution function F(y)

Question 3

How to calculate p y falls in interval a b when have density function

Answer 3

Take the integreal from a to b of the density function f(y)

Question 4

How to calculate the expected value of continues random variable

Answer 4

Take the integral over full range of f(y)*y or whatever is inside E[y]

Question 5

What is the expected value of g(y) a function of y

Answer 5

E(g(y)) = integral over full range of f(y)*g(y)

Question 6

Theorems of expected value E[y]

Answer 6

1. E(c) = c
2. E(cy) = cE(y)
3. if g1(y) , g2(y), g3(y) ect is function of y then
   E[g1(y)+ g2(y)+…] is equal to E(g1(y))+ E[g2(y)] ect

Question 7

How to calculate variance

Answer 7

E(y^2) - E[y]^2 where E[y] = mean

Question 8

What is the density function f(y) of a uniform distribution

Answer 8

f(y) = 1/upperbound - lowerbound where lower <= y <= upper

Question 9

What is the mean and variance of uniform distribution

Answer 9

mean is the mid value of range thus upperbound+lowerbound/2
Variance is defined as (upp - low)^2/ 12
Just integrate over the range and calculate E(y) for proof

Question 10

What is f(y) of a normally distubuted random variable

Answer 10

f(y) = 1/sigma root(2 pi) * e^(-(y-u)^2/2sigma^2)

Normal distribution has 2 parameters u and sigma
E[y] = u thus mean and v(Y) = sigma^2

Question 11

how to calculate p(y) for normal distribution

Answer 11

Use table of standard normal distrubution. Convert other normal distrubution to standard by z = y-u/sigma

Question 12

define f(y) for a gamma distribution

Answer 12

Gamma function always positive with positive constants a and b (alpha and beta)

y^(a-1)*e^(-y/b)/b^a * gamma(a) y >=0

Where gamma(a) = intigral from 0 to inf of y^a-1 *e^-y

Question 13

Properties of gamma function

Answer 13

gamma(1) = 1 and gamma(n) = (n-1)! for n is integer

Question 14

What is the expected value and variance of a gamma distribution.

Answer 14

E(y) = ab (alpha beta) V(y) = ab^2

Question 15

define parameters of chi-square distribution

Answer 15

if Y is gamma distrubution with a = v/2 where v is positive integer and b = 2 the it is a chi-squar distrubution with v degrees of freedom

Question 16

distibution of an exponential distibution

Answer 16

This is a gamma function with a = 1
thus distibution is
e^(-y/b) / b

Question 17

What is E(y) and V(y) for a exponential distibution

Answer 17

E(y) = b and V(y)= b^2

Question 18

What is the density of a beta distibution

Answer 18

y^(a-1)*(1-y)^b-1/B(a,b) where a and b is alpha and beta respectivly y is between 0 and 1

B(a,b) = integral form 0 to 1 y^(a-1)(1-y)^b-1 and this is equal to gamma(a)gamma(b)/gamma(a+b)

If a and b are integers the probability can be directly integrated to get probability ranges

Question 19

E(y) and V(y) for beta distribution

Answer 19

E(y) = a/a+b V(y) = ab/(a+b)^2*(a+b+1)

Question 20

Tchebyceff theorom

Answer 20

P(|Y-u| >= k*sigma) <= 1/k^2