# Solved – Sample space for discrete random variables

Can discrete random variables be defined in a continuous sample space?

Continuous sample space is a non countable sample space!

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For simplicity, let us assume that by "continuous sample space" you mean the real line $$mathbb R$$.
The corresponding probability space is $$(Omega, mathcal F, P)$$ where $$mathcal F$$ is the $$sigma$$-algebra of events defined on $$Omega$$ and $$P$$ is the probability measure that specifies the probability of each event in the $$sigma$$-algebra. For the sample space $$mathbb R$$, one can take the $$sigma$$-algebra to be generated the intervals $$(-infty, x]$$ and use any continuous nondecreasing function $$F$$ with limiting values $$0$$ as $$xto -infty$$ and value $$1$$ as $$x to infty$$ to define the probability of $$(-infty, x]$$ to be $$F(x)$$.
If the highlighted paragraph above is gobbledygook to you, just ignore it, and concentrate on what follows next. It is straightforward to define a discrete random variable on this sample space $$Omega = mathbb R$$. For example, one could define a discrete random variable $$X$$ as one that maps the outcome $$omega in Omega$$ (this outcome is a just an ordinary real number) to $$1$$ if $$omega leq 0$$ and to $$0$$ if $$omega > 0$$. $$X$$ is then just a Bernoulli random variable with parameter $$p= P(omega leq 0)$$.
The gobbledygook in the highlighted paragraph above just says that it makes sense to talk about the probability that the outcome $$omega$$ is no larger than $$0$$, or more generally, the probability that the outcome $$omega$$ is no larger than a specified real number $$x$$, and that this probability $$P(omega leq x)$$ is just the value of the function $$F$$ at $$x$$. In whuber's answer, he has chosen $$F$$ to be $$Phi$$, the cumulative probability distribution function (CDF) of the standard normal random variable.