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- TL;DR Summary
- Is a particular variant of Roy's discrete normal distribution also possible?

In the article

A discrete normal (##dNormal##) variate, ##dX##, can be viewed as the

discrete concentration of the normal variate ##X## following ##N(\mu,\sigma)## when the corresponding probability mass function of ##dX## can be written as

$$\displaystyle p \left(x \right) \, = \, \Phi((x+1-\mu)/\sigma) - \Phi((x-\mu)/\sigma) \hspace{3ex} x \, = \, \dots, \, -1, \, 0, \, +1, \, \dots$$

where ##\Phi(x)## represents the cumulative distribution function of the normal deviate ##Z##.

My first question is, is there a closed form for the expectation and variance of ##X##?

My second question is, is a discrete normal (##dNormal##) variate of the form

$$\displaystyle p \left(x \right) \, = \, \Phi((x+1/2-\mu)/\sigma) - \Phi((x-1/2-\mu)/\sigma) \hspace{3ex} x \, = \, \dots, \, -1, \, 0, \, +1, \, \dots$$

also possible and what is then the expectation and variance of ##X##? I suspect that the expectation of ##X## is equal to ##\mu##. But I don't know if the variance is equal to ##\sigma^2##.

**A Discrete Normal Distribution**of Dilip Roy in the journal COMMUNICATION IN STATISTICS*Theory and methods*Vol. 32, no. 10, pp. 1871-1883, 2003 one can read:A discrete normal (##dNormal##) variate, ##dX##, can be viewed as the

discrete concentration of the normal variate ##X## following ##N(\mu,\sigma)## when the corresponding probability mass function of ##dX## can be written as

$$\displaystyle p \left(x \right) \, = \, \Phi((x+1-\mu)/\sigma) - \Phi((x-\mu)/\sigma) \hspace{3ex} x \, = \, \dots, \, -1, \, 0, \, +1, \, \dots$$

where ##\Phi(x)## represents the cumulative distribution function of the normal deviate ##Z##.

My first question is, is there a closed form for the expectation and variance of ##X##?

My second question is, is a discrete normal (##dNormal##) variate of the form

$$\displaystyle p \left(x \right) \, = \, \Phi((x+1/2-\mu)/\sigma) - \Phi((x-1/2-\mu)/\sigma) \hspace{3ex} x \, = \, \dots, \, -1, \, 0, \, +1, \, \dots$$

also possible and what is then the expectation and variance of ##X##? I suspect that the expectation of ##X## is equal to ##\mu##. But I don't know if the variance is equal to ##\sigma^2##.

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