1. **Problem Statement:** Determine the truth value of each statement about characteristic functions and moments of a random variable. 2. **Recall definitions and facts:** - The characteristic function of a random variable $X$ is defined as $\varphi_X(t) = \mathbb{E}[e^{itX}]$. - Moments of $X$ relate to derivatives of $\varphi_X(t)$ at $t=0$ if they exist. 3. **Analyze each statement:** **a.** If $\varphi_X(t)$ is differentiable at $t=0$, then the first moment $\mathbb{E}[X]$ exists. - The first derivative at zero is $\varphi_X'(0) = i\mathbb{E}[X]$ if $\mathbb{E}[X]$ exists. - Differentiability at zero implies $\mathbb{E}[X]$ exists. - **True**. **b.** If the second derivative of $\varphi_X(t)$ exists at $t=0$, then the variance of $X$ exists. - The second derivative at zero is $\varphi_X''(0) = i^2 \mathbb{E}[X^2] = -\mathbb{E}[X^2]$ if $\mathbb{E}[X^2]$ exists. - Existence of second derivative implies $\mathbb{E}[X^2]$ exists, so variance exists. - **True**. **c.** The characteristic function always exists regardless of moments. - By definition, $\varphi_X(t)$ exists for all $t$ since $|e^{itX}|=1$. - Moments may not exist, but characteristic function does. - **True**. **d.** Differentiability of $\varphi_X(t)$ at $t=0$ does not guarantee existence of all higher moments. - Existence of $n$th moment requires $n$th derivative at zero. - Differentiability at zero only guarantees first moment. - **True**. **e.** Characteristic function uniquely determines the distribution. - By Lévy's continuity theorem, characteristic function uniquely identifies distribution. - **True**. 4. **Final answers:** - a: True - b: True - c: True - d: True - e: True All statements are true based on properties of characteristic functions and moments.