# Probability - transformation of random variable

• May 4th 2011, 02:06 PM
dizzle1518
Probability - transformation of random variable
In an analog to digital conversion and analog waveform is sampled, quantized and coded. A quantized function is a function that assigns to each sample value x a value y from a generally finite set of predetermined values. Consider the quantized defined by g(x)=[x]+1, where [x] denotes the greatest integer less than or equal to x. Suppose that x has a standard normal distribution and pit Y=g(x). Specify the distribution of Y. Ignore values of Y for which the probability is essentially zero.

Going by how the book taught it I would start this problem by computing the inverse of g(x). However this function has no inverse. Any suggestions how to proceed?
• May 4th 2011, 11:27 PM
Moo
Hello,

No you don't need.
Y has integer values.

So what's the probability for Y to be equal to an integer k ?

P(Y=k)=P([x]+1=k)=P([x]=k-1)

For [x] to be equal to k-1, that means that x is between k-1 and k (definition).

So P(Y=k)=P(k-1 < x < k)

There's no way of simplifying it, but this defines Y's distribution.
• May 5th 2011, 05:33 AM
dizzle1518
Quote:

Originally Posted by Moo
Hello,

No you don't need.
Y has integer values.

So what's the probability for Y to be equal to an integer k ?

P(Y=k)=P([x]+1=k)=P([x]=k-1)

For [x] to be equal to k-1, that means that x is between k-1 and k (definition).

So P(Y=k)=P(k-1 < x < k)

There's no way of simplifying it, but this defines Y's distribution.

going off what you wrote P(Y<=k)=P(x<k). since x follows a standard normal distribution we can write the distribution of Y as Fy=Phi (k) where Phi is the cumulative distribution function of the normal random variable?