Consider k continuous-type distributions with the following characteristics: pdf f i (x), mean i , and

Consider k continuous-type distributions with the following characteristics: pdf f_i(x), mean μ_i, and variance σ²_i , i = 1, 2, . . . , k. If c_i ≥ 0, i = 1, 2, . . . , k, and c₁+c₂+· · ·+c_k = 1, show that the mean and the variance of the distribution having pdf c₁f₁(x) + · · · + c_kf_k(x) are μ = Σ^k_i=1 c_iμ_i and σ² = Σ^k_i=1 c_i[σ²_i + (μ_i − μ)²], respectively.