# Math Help - Find distribution function of a mixed variable

1. ## Find distribution function of a mixed variable

X follows U[0,1]
Y follows discrete uniform {0,1,..,n-1}
Z=X+Y.
distribution of Z?

2. Are they independent ?

If so, let k be an positive integer. Consider a z in $[0,n)$. Let z=x+y, where x is the decimal part of z, and y its integer part.

$P(y\leq Z\leq x+y)=P(Y=y,X\leq x)=\frac xn$

So the probability that Z is in [k,k+1)=1/n (letting x=1)

So $P(Z\leq z)=P(0\leq Z

This is the cdf of a uniform distribution over $[0,n]$