cumulative distribution function of a uniform distribution

#1
I just need some help on the question below.

Bacteria are grown in a dish for a length of time (in hours) T which is a random variable with a uniform distribution over the range 5 to 6; that is, it has the pdf

F(t) = {1, 5<= t <=6} {0, otherwise}

the number of bacteria Y, in the dish is given by Y = e^(cT) where c is a positive constant.

(i) Write down the cdf of T and hence find the cdf of Y
(ii) Sketch both these cdfs
(iii) Differentiate the cdf of Y to find and sketch the pdf of Y
(iv) Also use the method of transformations to find the pdf of Y directly from the pdf of T.

(ii) and (iii) I should be able to do but i need help with (i) and (iv).

Thanks
 

BGM

TS Contributor
#2
By definition

\( F_T(t) = \Pr\{T \leq t\} = \int_{-\infty}^t f_T(x)dx \)

Of course you have to split the trivial cases when \( t \) is too small or too large as \( T \) has bounded support.