7 - continuous distribution

Question 1

probability density function

Answer 1

f(x) the function where the area under shows the probability of the crv being between those values
the total area = 1
f(x) is aways >= 0

Question 2

P(a<X<b) =

Answer 2

= ∫ f(x) dx
from b to a

Question 3

linear transformation

Answer 3

E(aX+c) = aE(X) +c
Var(aX+c) = a^2Var(X)

Question 4

E(X)

Answer 4

∫x f(x) dx
from b to a

Question 5

median

Answer 5

∫ f(x) from m to a = 1/2

Question 6

general transformations eg E(g(x))

Answer 6

= ∫g(x)f(x) dx
form b to a

Question 7

cumulative distribution function

Answer 7

the probabitlity of being less than or = to a value
F(x) = P(X<=x) = ∫ f(x) dx from x to a
f(x) = d/dx F(x)

Question 8

piecewise defined pdf

Answer 8

eg f(x) = x^2 for 0-2
and = x^3 for 2-4
just split and integrate each part separately

Question 9

continuous uniform distribution

Answer 9

any equally sized part of the domain has an equal prop of occurring
the pdf is constant and is chosen so the area under the graph = 1
graph is a rectangle
if X is uniform continuous from a-b
then f(x) = 1/b-a from a-b

Question 10

Var(X) and E(X) for uniform

Answer 10

E(X) = a+b/2
Var(X) = (b-a)^2 /12

Question 11

exponential distribution

Answer 11

X~Exp(λ)
f(x) = λe^-λx - formula book
E(X) = 1/λ
Var(X) = 1/λ²
models the waiting interval in a poisson type process - the interval between events
is memory less - doesn’t matter when the last event occurred

Question 12

if X~Exp(λ) then F(x)

Answer 12

= 1-e^-λx

Question 13

exponential + poisson

Answer 13

if A~Po(6) in 1hour
then B~Exp(6) also in one hour

Question 14

distributions of functions of crv eg if Y = √X then whats the pdf of Y

Answer 14

if F(x) = 1/16 x² and Y = √X find pdf of Y?
- G(Y) = P(Y<=y) = P(√X <=y) = P(X<=y²)
P(X<=x) = 1/16 x² then P(X<=y²) = 1/16 y^4
g(y) = d/dy G(Y) = d/dy 1/16 y^4= 1/4y^3
if x is from 0-4 y^2 is from 0-4 so y is from 0-2

Question 15

goodness of fit test of continuous

Answer 15

• find the expected frequency by doing the prop * the total number
• define H0 and H1 - H0 is that they have the distribution H1 is they dont
• do X^2 - sum (O-E)^2/E
• degrees of freedom = number of columns -1 and -1 again if you estimate any parameters
• compare with the value in the table
• if value < table then insufficient evidence to reject H0
• if value > table then sufficient evidence to reject H0
• conclusion in context