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Question Number 102085 Answers: 0 Comments: 0

Question Number 102080 Answers: 0 Comments: 0

A random variable, X, has a Gamma distribution with parameters α and β, (α, β>0). The p.d.f has the form f(x)=(1/(Γ(α)β^α ))x^(n−1) e^(−x/β) , for x>0 , Γ(α)=(1/β^α )∫_0 ^∞ x^(α−1) e^(−x) dx a\ Show that the Gamma density is a proper p.d.f. b\Find the mean, variance, and moment-generating function of the Gamma distribution. c\Find the fourth moment using the definition of moments.

$$\mathrm{A}\:\mathrm{random}\:\mathrm{variable},\:\mathrm{X},\:\mathrm{has}\:\mathrm{a}\:\mathrm{Gamma}\:\mathrm{distribution}\:\mathrm{with} \\ $$$$\mathrm{parameters}\:\alpha\:\mathrm{and}\:\beta,\:\left(\alpha,\:\beta>\mathrm{0}\right).\:\mathrm{The}\:\mathrm{p}.\mathrm{d}.\mathrm{f}\:\mathrm{has}\:\mathrm{the}\:\mathrm{form} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\Gamma\left(\alpha\right)\beta^{\alpha} }\mathrm{x}^{\mathrm{n}−\mathrm{1}} \mathrm{e}^{−\mathrm{x}/\beta} ,\:\mathrm{for}\:\mathrm{x}>\mathrm{0}\:\:,\:\:\Gamma\left(\alpha\right)=\frac{\mathrm{1}}{\beta^{\alpha} }\int_{\mathrm{0}} ^{\infty} \mathrm{x}^{\alpha−\mathrm{1}} \mathrm{e}^{−\mathrm{x}} \mathrm{dx} \\ $$$$\mathrm{a}\backslash\:\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{Gamma}\:\mathrm{density}\:\mathrm{is}\:\mathrm{a}\:\mathrm{proper}\:\mathrm{p}.\mathrm{d}.\mathrm{f}. \\ $$$$\mathrm{b}\backslash\mathrm{Find}\:\mathrm{the}\:\mathrm{mean},\:\mathrm{variance},\:\mathrm{and}\:\mathrm{moment}-\mathrm{generating}\:\mathrm{function}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{Gamma}\:\mathrm{distribution}. \\ $$$$\mathrm{c}\backslash\mathrm{Find}\:\mathrm{the}\:\mathrm{fourth}\:\mathrm{moment}\:\mathrm{using}\:\mathrm{the}\:\mathrm{definition}\:\mathrm{of}\:\mathrm{moments}. \\ $$

Question Number 102076 Answers: 0 Comments: 4

A car is currently valued at $70350.00. If it loses 12% of its value at the beginning of each year, a) find its value after three and half years. b) find the depreciation after three years

$$\mathrm{A}\:\mathrm{car}\:\mathrm{is}\:\mathrm{currently}\:\mathrm{valued}\:\mathrm{at}\:\$\mathrm{70350}.\mathrm{00}. \\ $$$$\mathrm{If}\:\mathrm{it}\:\mathrm{loses}\:\mathrm{12\%}\:\mathrm{of}\:\mathrm{its}\:\mathrm{value}\:\mathrm{at}\:\mathrm{the}\:\mathrm{beginning} \\ $$$$\mathrm{of}\:\mathrm{each}\:\mathrm{year}, \\ $$$$\left.\mathrm{a}\right)\:\mathrm{find}\:\mathrm{its}\:\mathrm{value}\:\mathrm{after}\:\mathrm{three}\:\mathrm{and}\:\mathrm{half}\:\mathrm{years}. \\ $$$$\left.\mathrm{b}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{depreciation}\:\mathrm{after}\:\mathrm{three}\:\mathrm{years} \\ $$

Question Number 101794 Answers: 1 Comments: 0