Rvs, PMFS, PDFS

Question 1

What is sample space

Answer 1

Sample space is a set of all possible outcomes.

Question 2

When is X a continuous RV

Answer 2

X is a continuous RV when there’s a continuous function fx such that:
P(a <= x <= b) = integral b down to a fx dx

Question 3

If fx is a PDF, what must it have

Answer 3

If fx is a PDF, it must have:
Fx(x)>=0 for all x in R
Integral infinity down to -infinity fx dx =1

Question 4

What is P(X=a) for a continuous RV

Answer 4

P(X=a) for a continuous RV is:

Integral a down to a fx(x) dx =0

Question 5

What is PDF of RV X if X~U(a,b)

Answer 5

PDF of RV X if X~U(a,b) is:

Fx(x)= 1/b-a for a < x < b and 0 otherwise

Question 6

What is the CDF of RV X

Answer 6

CDF of RV C is:

Fx(x) = P(X<=x) (applies to both continuous and discrete)

Question 7

What is the relationship between CDF and PDF (only holds for continuous)

Answer 7

Relationship between CEF and PDF is:
Fx(x)= P(X<=x)= integral x down to -infinity fx(u)du
So fx(x)=derivative of Fx(x)

Question 8

What is the transformation formula

Answer 8

Transformation formula is:
Let X be a continuous RV with P(a < X < b)=1 and g is a continuous, strictly increasing and differentiable. Then Y = g(X) is a continuous RV with PDF
fy = fx(y)* dx/dy