"The random variable X has uniform distribution on range 0 to 1, write down its pdf":

Now this seems a bit easy but I never learnt the basics before going onto the advanced stuff so I am not sure, I had to look it up. Is it correct that it is 1 for x between 0 and 1 and 0 elsewhere?

"Show that the c.d.f is 0 for x is less than or equal to 0, x for x between 0 and 1, and 1 for x is greater equal to 1":

Now the only thing I can think of is integrating the 1 (0<x<1) to get the x for that part, is that all you can do to show it? Should the fact f(x) = 1 for x>1 be a given?

"Hence find the c.d.f. of the random variable Y = -2lnx and find the pd.f and variance of Y identifiying the distribution of Y"

Ok so what do you sub in here? Y or -2lnx?