Suppose X has the Uniform 01 distribution Find the median of

Suppose X has the Uniform (0,1) distribution. Find the median of the distribution of e to the power of X correct to 2 decimals.

f(x) = 1 , 0 < x < 1

Then it can be written as,

y = e^x

P( e^x < k)

= P( x < ln (k) )

= ln(k)

Thus, the distribution function of

y= e^x is ln(x)
F(x) = ln (x)
F\'(x) = 1/x
f(y) = 1/ y

That is

when x=0, y=e^0 =1
when x=1, y=e^1 = e

So, the distribution of y=e^x is
f(y) = 1/y , 1 < y < e

The median is:

[1,m] 1/y dy = 0.5
1/y dy = ln(y)
ln(m)-ln(1) = 0.5
ln(m) = 0.5
m = e^0.5 =2.71^0.5=1.65