Discussion
Suppose that X is a random variable with c.d.f. F. Then, the distribution F is not symmetric around its median when :
Answer: C
The distribution is not symmetric about its median when F is exponential.
The exponential density is \(\lambda e^{-\lambda(x - \theta)}\) for \(x \ge \theta\) and 0 below \(\theta\): it has a long right tail and no left tail at all, so it cannot be a mirror image about its median \(\theta + \frac{\ln 2}{\lambda}\).
The others are symmetric. Binomial(n, ½) has \(P(X = k) = P(X = n - k)\), symmetric about \(\frac{n}{2}\), its median. The uniform on (a, b) is symmetric about \(\frac{a + b}{2}\), and the Cauchy is symmetric about its location parameter \(\theta\), which is its median.
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