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?
2026-04-03 16:44:07.1775234647
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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.