Let $X_k$ ($1 \leqslant k \leqslant n$) be real valued random variables with corresponding probability density function $f_{X_k}$ ($1 \leqslant k \leqslant n$), respectively. Define the joint density function $f_{X_1,\cdots,X_n}(x_1, \cdots, x_n) =C_1f_{X_1}(x_1)+ \cdots +C_nf_{X_n}(x_n)$
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(I) Find a condition on the n tuple $C = (C_1, · · · ,C_n)$.
This is what I did : By the definition of joint density and its imposed condition, we know that: $$\int_{-\infty}^\infty \dots \int_{-\infty}^\infty f(x_1, \cdots, x_k)dx_1,\cdots,dx_k=1$$
Since the joint density function $f_{X_1,\cdots,X_n}(x_1, \cdots, x_n) =C_1f_{X_1}(x_1)+ \cdots +C_nf_{X_n}(x_n)$ is given, then:
\begin{align}& \Longrightarrow \int_{-\infty}^\infty \dots \int_{-\infty}^\infty (C_1f_{X_1}(x_1)+ \cdots +C_nf_{X_n}(x_n))dx_1,\cdots,dx_n=1\\ & \Longrightarrow \int_{-\infty}^\infty C_1f_{X_1}(x_1) dx_1+ \cdots +\int_{-\infty}^\infty C_nf_{X_n}(x_n))dx_n=1\\ & \Longrightarrow C_1\int_{-\infty}^\infty f_{X_1}(x_1) +\cdots + C_n \int_{-\infty}^\infty f_{X_n}(x_n) dx_n=1\\ & \Longrightarrow \boxed{C_1\cdots + C_n =1, \quad C_{1,\cdots n} \in \mathbb{R}}\\ \end{align} Is it right?