I think we cannot convert discrete probability distribution into continuous probability distribution

We may want to convert it in case of having poison distribution of having discrete values of x as in x is the numbers of machines with errors (could be 1,2,3,4 or 5). But in continuous you may want to say x is equal to 1 or greater than. Meaning one has to work on 1 and above till the probability for the continuous variable becomes the level we can say it is low.

No, there is no "conversion" between discrete and continuous random variables (RV). It's only possible in some cases to make a discrete RV so fine-grained that it can be approximated by a continuous RV. One example is the binomial distribution, Bin(n,p). For n -> Inf and p=µ/n the distribution will be approximate a normal distribution with mean µ (and s²=0).

Dear All, let me add my 4 pennies:

1. EVERY continuous distribution function [df] can be approximated arbitrarily close by a df of a classical probability; namely:

the classical probability distribution concentrated on quantiles q_{i/n} of the given probability distibution say F, (i.e. quantiles of order i/n, for i=1,2,...,n-1 with probability 1/(n-1) each) possesses the uniform distance from the original F EQUAL to 1/(n-1) ;

2. A well known example of the inverse way is that:

the df of the sum of n iid rv-s with finite variance v and mean zero divided by radical of n approaches the df of the normal distribution with variance v, at every point as n-> oo, whenever v>0;

3. Bin(n, µ/n ) converges to the Poisson probability distribution with parameter µ, as n->oo, whenever µ is positive;

4. And some more general statement:

for ANY sequence of probability distributions concentrated on integers the limit probability distribution is also concentrated on integers, whenever the limit exists.

Let (Ω,F,P) be a probability space. Let X:Ω→R be a random variable defined on this probability space. Suppose that X is a discrete random variable. If g:R→R is any function, then g(X), which is indeed a composite function g∘X:Ω→R defined by goX(ω):=g(X(ω)) for all ω∈Ω, is called a transformation of the random variable X.

Now, the question is, can g(X) be a continuous random variable given that X is discrete. In other words, can the distribution of g(X) be a continuous one?

A discrete random variable is a one for which the range is finite or countably-infinite. A continuous random variable is a one with uncountably-infinite range. Since X is a discrete random variable, the range of X, which is the set {X(ω):ω∈Ω}, is finite or countable.

The original question thus reduces to, can the range of the random variable g(X) be uncountable ? Do you think it is plausible?

Certainly not. The range of g(X) is obtained by mapping each element of the range of X, which is the set {X(ω):ω∈Ω}, according to the function g. Since g is a function, each element of the set {X(ω):ω∈Ω} can be mapped to one, and only one, element. Thus the range of g(X) cannot become bigger than the range of X. In other words, the range of the random variable g(X) cannot be uncountably infinite. Therefore g(X) cannot be a continuous random variable.

There are some ways: approaching some parameters to zero, infinity (like normal), a couple of them, or a combination of them (like Poisson as a limit of binomial).

The other method is defining a continuous variable based on a discrete (like exponential as waiting time for a Poisson).

To Jochen Wilhelm, Adeyeye John and Joachim Domsta:

What exactly does the phrase mean ""conversion" between discrete and continuous random variables (RV)?"

To Romeo Mestrovic

Conversion in this context means: can we turn/ transform a discrete distribution to continuous distribution?

To Adeyeye John, Jochen Wilhelm, and Joachim Domsta:

I no longer have time to participate in this discussion. Namely, your phrase

“turn/transform a discrete distribution to continuous distribution“ is quite inaccurate in Statistics and Probability Theory, such as the term "conversion" given in the answer posted by Jochen Wilhelm.

Dear All,

The issue is not that simple that we can neglect some possibilities in positive understanding of the notion "turn/transform a discrete" pd into a continuous pd*).

For example, one might be interested in some operation that assigns to any discrete pd a continuous one NOT necessarily by an outer function [this case is discribed by Abolfazl Ghoodjani with the main conclusion that no discrete pf can be transfomed into a continuous one in this way].

But there are other operations like smoothing via a convolution, say with a normal pd of mean 0 and the variance value somehow adapted to the average distance between consecutive points of the discrete pd.

As a particular example let me point out that the geometric pd

{(0,1-p), (1, p(1-p)), (2, p^2 (1-p), . . . } , with 0

Every question related to a mathematical problem (a conjecture,…) must be strictly specified (defined), in order to possibly get adequate answers (solutions).

YES, dear Romeo! I agree with you if the problem is on mathematics.

But my opinion is different if the aim of the question to be answered is oriented toward explaining which mathematical description might be appropriate. Especialy for a problems born during studying some real phenomena, where mathematics seems to be a valuable tool of its analysis.

Also - as in the current case - someone is familiar with absolutely continous pd-s more than with the discrete ps-s; OR if one likes presentation in terms of density function more than in terms of discrete pd-s, etc.

Best regards, Joachim

Thanks @ Romeo and Joachim. I must say that I've tapped from your wealth of

knowledge.

God bless you all.

Thank you Adeyeye John for kind words. Best regards, Joachim Domsta

Thanks @ Mehdi Mesrizadeh

In my understanding, for instance: the Poisson model (Discrete) turns into a (continuous) known as the Zero Truncated Poisson model by making some kind of algebra and ready to interact with continuous models in different ways.

Muhammad Zeshan Arshad

Your comment is very interesting. Can you explain please, what is the way that turns Poisson discrete into Zero truncated Poisson.

https://link.springer.com/content/pdf/10.1007/s40745-019-00201-y.pdf

Muhammad Zeshan Arshad

Thank you very much,

Joachim Domsta