X follows U[0,1]

Y follows discrete uniform {0,1,..,n-1}

Z=X+Y.

distribution of Z?

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- Jun 10th 2010, 01:24 AMDootiFind 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? - Jun 10th 2010, 03:37 AMMoo
Are they independent ?

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

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

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

So $\displaystyle P(Z\leq z)=P(0\leq Z<y)+P(y\leq Z\leq x+y)=\frac yn+\frac xn=\frac{x+y}{n}=\frac zn$

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