3. Questions

Question 1

If X has the pdf f(x)=2cx if x is from 0 to 2 and f(x)=0 otherwise. How do I determine the value of c?

Answer 1

The sum of the PDF between 0 and 2 must equal 1 so integrate f(x) to find the area under the curve and set it equal to 1

Question 2

When given the pdf how do you find the probability of P(1<x<2)

Answer 2

Integrate the pdf with the limits of 2 and 1.

Question 3

When given the joint pdf how do you find the probability of P(0<X<0.5, 0.5<y<1)

Answer 3

First integrate the joint pdf wrt dx. Sub in the values of x to give a function just in terms of y. Then integrate this function wrt dy and sub in y values to give answer

Question 4

If you have the joint pdf and 0<x<1, 0<y<2. How do you find P(X>0.5)

Answer 4

First calculate the marginal PDF of x by integrating the pdf wrt dy and subbing in y values 2 and 0. Then integrate the marginal PDF wrt dx and sub in values 1 and 0.5 to get answer

Question 5

How do you work out the density function of Y given X=x?

Answer 5

f(y|x)= f(x,y)/f(x)

In words the density function is the joint pdf divided by the marginal pdf of x

Question 6

Given a joint pdf how do you find P(0<y<0.5, | x=0.5)

Answer 6

Calculate the density function of y given X=x. Then calculate the conditional density of Y at x=0.5 by subbing in x=0.5. Then integrate this function and sub in y values