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

Continuous sample space is a non countable sample space!

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#### Best Answer

For simplicity, let us assume that by "continuous sample space" you mean the real line $mathbb R$.

The corresponding

probability spaceis $(Omega, mathcal F, P)$ 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.

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