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Math Help - Find distribution function of a mixed variable

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    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?
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    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<y)+P(y\leq Z\leq x+y)=\frac yn+\frac xn=\frac{x+y}{n}=\frac zn

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