- #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 < e

^{y})

I know that X is uniform over [0,1] so this is equal to

∫1.dy between e

^{y}and 0, where e

^{y}≤ 1

This gives me that the cumulative distribution function = e

^{y}

Which tells me that the probability density function is also e

^{y}

Now what I am not sure of is what must y lie between, is this for y between e

^{b}and e

^{a}?