# PDF Random Variable Statement

Is it true that if the probability density function of a continuous random variable is an even function, then the continuous random variable is symmetric?

• You get that this is just saying an even function is symmetric around zero? The only connection to probability is the definition of a continuous random variable being symmetric (which is that its pdf is symmetric at some point - which in this case happens to be zero). Feb 12 at 13:28

Yes, this is true. Consider a probability density function $$p(x)$$ that is even:
By definition, the probability density function is symmetric if there exists $$x_0$$ s.t $$p(x_0+δ)=p(x_0-δ)$$ for all real numbers δ. But take $$x_0=0$$. Then it follows immediately that if p is even then $$p(δ)=p(-δ)$$ for all real $$δ$$. So clearly the probability density function is symmetric.
• Strictly speaking you showed that for $x_0\not=0$ the PDF is not even. Feb 12 at 10:52
• Sorry I got it wrong. All correct: if $p$ is even then RV is symmetric. I had the converse implication in mind. Feb 12 at 11:01