Question

Let \(X\) be a random variable with the following probability density function
\[ f(x)= \begin{cases} 4x^2e^{-2x} & \text{if } x>0 \\ 0 & \text{otherwise.} \end{cases} \]
If \(Y=\ln X\), then which of the following statements is correct?

• A. \(E(Y)\) is not finite
• B. \(E(Y)=\ln\dfrac{3}{2}\)
• C. \(E(Y)>\ln\dfrac{3}{2}\)
• D. \(E(Y)<\ln\dfrac{3}{2}\)

Hint:

\(X\) follows a Gamma(3,2) distribution with \(E(X)=3/2\). Since \(\ln\) is strictly concave, Jensen's inequality gives \(E(\ln X)<\ln(E(X))\).

Answer 1

The Correct Option is D

Step 1: Identify the distribution of X.
The given density is \[ f(x)=4x^2e^{-2x}, \qquad x>0. \] Compare this with the Gamma density with shape parameter \(\alpha\) and rate parameter \(\beta\): \[ f(x)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}. \] With \(\alpha=3\) and \(\beta=2\), \[ \frac{\beta^{\alpha}}{\Gamma(\alpha)}=\frac{2^3}{\Gamma(3)}=\frac{8}{2!}=\frac{8}{2}=4, \] which matches exactly. So \(X\sim Gamma(\alpha=3,\beta=2)\).

Step 2: Find \(E(X)\).
For a Gamma random variable with shape \(\alpha\) and rate \(\beta\), \[ E(X)=\frac{\alpha}{\beta}=\frac{3}{2}. \]

Step 3: Recall the key inequality for a concave function.
The natural logarithm \(\ln(\cdot)\) is a strictly concave function on \((0,\infty)\). Jensen's inequality states that for a strictly concave function \(g\) and a non-degenerate random variable \(X\), \[ E[g(X)]<g(E[X]). \] The inequality is strict, not just \(\le\), whenever \(X\) is not a constant, which is true here since \(X\) has a continuous density.

Step 4: Apply Jensen's inequality to \(Y=\ln X\).
Taking \(g(x)=\ln x\), \[ E(Y)=E[\ln X]<\ln[E(X)]=\ln\!\left(\frac{3}{2}\right). \]

Step 5: Confirm \(E(Y)\) is finite.
Since \(X\) has a Gamma density with shape \(3\), the moment \(E[\ln X]\) exists and is finite: the exponential decay dominates the logarithmic growth near infinity, and near \(0\) the term \(x^2\ln x\to0\). This rules out option (A), which claims \(E(Y)\) is not finite.

Step 6: Check the remaining options against Step 4.

(B) \(E(Y)=\ln(3/2)\): This would need equality in Jensen's inequality, which only happens when \(X\) is a constant. Since \(X\) is genuinely random here, this is false.

(C) \(E(Y)>\ln(3/2)\): This is the reverse of what Jensen's inequality gives for a concave function, so it is false.

(D) \(E(Y)<\ln(3/2)\): This matches exactly what we derived in Step 4.

Final Answer:
Because the logarithm is strictly concave and \(X\) is not degenerate, \[ \boxed{E(Y)<\ln\frac{3}{2}} \]