Calculate $P(A = \frac 34)$, given the PDF of $A$

Question

$A$ is absolutely continuous random variable. Given a density function $fa(a)$ I have to calculate $P(A = \frac 34)$.

I tried to just calculate the integral with both lower and upper limit $\frac 34$ and then $f(a)$ da but that just gives zero. What would be correct to do?

Answer 1

You are correct.

If $dF/dx = f$ then $$P(p \le X \le q) =\int_{p}^{q}f(x)\,dx=\left[F(x)\right]_{p}^{q}=F(q)-F(p)$$ If $p=q$ then this equals zero ($p=q=3/4$ in your case.)

For a discussion about the intuition behind this, see this question