What is the distribution function of a random variable X which has a constant fixed value of 2?

Question

I remember reading somewhere that it is not possible for a continuous variable to take on an exact value so would the answer be 0?

Answer 1

You can construe such a variable as either continuous or discrete. The probability density would not be a function but a measure, the Dirac Delta $\delta(x-2)$. Alternatively, the probability mass function is a Kronecker Delta, $\delta_{x2}$. See here

Answer 2

Find the CDF $F$ of $X$:

$$F(x) = \text{Pr}(X \le x)$$

• If $x < 2$, then $F(x) = \text{Pr}(\varnothing) = 0$.

• If $x \ge 2$, then $F(x) = 1$.

The PDF $f$ of $X$ is $$f(x) = \frac{d}{dx} F(x)$$

wherever defined.

Which is $0$ everywhere obviously.