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Math Help - Probability - transformation of random variable

  1. #1
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    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?
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  2. #2
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    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.
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  3. #3
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    Quote Originally Posted by Moo View Post
    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?
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