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Transformation of the uniform distribution

  • Thread starter stukbv
  • Start date
  • #1
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Homework Statement



I am told that X is a random variable with uniform distribution over [0,1]
I need to find the mean and variance of log(X)

2. The attempt at a solution
I assume I must find the pdf of log(X) so I did this as follows;

Let Y=log(X)
Then to find the cumulative distribution function I considered;
P(Y≤ y) = P(log(X)≤ y) = P( X < ey)

I know that X is uniform over [0,1] so this is equal to
∫1.dy between ey and 0, where ey≤ 1

This gives me that the cumulative distribution function = ey
Which tells me that the probability density function is also ey

Now what I am not sure of is what must y lie between, is this for y between eb and ea ?
 

Answers and Replies

  • #2
Ray Vickson
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Homework Statement



I am told that X is a random variable with uniform distribution over [0,1]
I need to find the mean and variance of log(X)

2. The attempt at a solution
I assume I must find the pdf of log(X) so I did this as follows;

Let Y=log(X)
Then to find the cumulative distribution function I considered;
P(Y≤ y) = P(log(X)≤ y) = P( X < ey)

I know that X is uniform over [0,1] so this is equal to
∫1.dy between ey and 0, where ey≤ 1

This gives me that the cumulative distribution function = ey
Which tells me that the probability density function is also ey

Now what I am not sure of is what must y lie between, is this for y between eb and ea ?
For X between 0 and 1, what is the range of log(X)?

Anyway, to get the mean and variance of Y = log(X), you don't need the distribution of Y, although getting it is certainly one way of doing the problem.

RGV
 
  • #3
118
0
I think my lecturer wants me to do it this way. So y must be between - infinity and 0?
Is my pdf correct now?
 

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