Statistics PDF and change of variable

[Larrytsai]

If the probability density of X is given by

f(x) =

{2(1 − x) for 0 ≤ x ≤ 1
{0 otherwise


(a) Find the probability density function of Y1 = 2X − 1.

I do not know how to start this problem can someone please help.
Is there a formula that I am missing from my notes to solve this problem?

 

[LCKurtz]

Start by finding the cumulative distribution function of Y:

P(Y ≤ y) = P(2X-1 ≤ y) = P(X ≤ ?)

Figure that out for various y and use the fact that the density function is the derivative of the distribution function.

 

[Bacle]

To add to what LC said:

If you know the distribution/density of a variable X, what can you say about

the density/distribution of f(X)?

 

[Larrytsai]

LCKurtz:

so P(X≤ (Y+1/2))

Bacle:

that means we can say f(X) gives us the probability of X?

 

[LCKurtz]

Start by finding the cumulative distribution function of Y:

P(Y ≤ y) = P(2X-1 ≤ y) = P(X ≤ ?)

Figure that out for various y and use the fact that the density function is the derivative of the distribution function.

LCKurtz:

so P(X≤ (Y+1/2))

That isn't correct as you have written it. You need proper parentheses and, to be pedantic about it, a lower case y. Then use the density of X to calculate the probability.

 

[Bacle]

Larry:

Look at the section on Functions of a Random Variable" :

http://en.wikipedia.org/wiki/Random_variables

which generalizes the fact , in your example, of Y being a function of X.

 

[aliafi90]

can anyone explain the steps towards the solution? I'm really confused here too,thank you