Discussion
Suppose that X is a random variable with cumulative distribution function F, probability density function f, and moment generating function M. If M satisfies the condition :
\[ e^{-5t} M(t) = e^{5t} M(-t), \quad \forall\, t \ge 0, \]
then which of the following statements is FALSE ?
Answer: B
The FALSE statement is \(f(x - 5) = f(5 - x)\ \forall\, x\).
Rewrite the condition: \(e^{-5t}M(t) = E\,e^{t(X - 5)}\) is the mgf of \(X - 5\), and \(e^{5t}M(-t) = E\,e^{t(5 - X)}\) is the mgf of \(5 - X\). Equal mgfs mean \(X - 5\) and \(5 - X\) have the same distribution, so X is symmetric about 5.
Symmetry about 5 gives \(f(5 + x) = f(5 - x)\) (option A), makes 5 the median (option C), and is exactly option D.
Option B, \(f(x - 5) = f(5 - x)\), says f is an even function, symmetric about 0, which is a different property and false in general (for example for N(5, 1)).
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