need help with determining cumulative distribution functions

Question

The problem is:

The amount of data, X, arriving at a communication channel, consisting of one main channel and two overload channels, is modelled by a continuous random variable which is uniformly distributed on the interval [0, 6]. The capacity of the main channel is equal to d1 > 0. If X ≤ d1, all the arriving data is transmitted over this main channel. However, if X > d1, only an amount of d1 can be transmitted over the main channel and the remaining part X − d1 should be transmitted over the overload channels. The capacity of the first overload channel is equal to d2 − d1 > 0. Hence, if d1 < X ≤ d2, the remaining part of the data can (and will) be completely transmitted over this first overload channel. However, if X > d2, also the second overload channel has to be used. In this case d1 is transmitted over the main channel, d2 −d1 is transmitted over the first overload channel and X −d2 is transmitted over the second overload channel. Let Y0, Y1 and Y2 denote the amount of data transmitted over the main channel, the first overload channel and the second overload channel, respectively.

Give the cumulative distribution functions of the random variables Y0, Y1 and Y2 and discuss whether Y0, Y1 and Y2 are discrete, continuous or mixed random variables.

I know the pdf of X is 1/6 and the cdf is x/6, but I don't know what to do after that with Y0 etc. Thanks for your time!

Answer 1

The CDF of Y0 is: 0 y0 < 0

y0/6 0 <= y0 < d1

1 y0 >= d1

The CDF of Y1 is: 0 y1 < 0

d1/6 y1 = 0

d1/6 + y1/6 0 < y1 < d2 - d1

1 y1 >= d2 - d1

The CDF of Y2 is: 0 y2 < 0

d2/6 y2 = 0

d2/6 + y2/6 0 <= y2 < 6 - d2

1 y2 >= d2 - d1

These are all mixed random variables. Note the discontinuities in the CDF.