Answer in Statistics and Probability for PRS #242554

Given that u=“\chi”2m and v=“\chi” n2 further let u and v be independent variables obtain the probability density of F=(U/M)/(V/N) where m and n are the df for U and V

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Answer in Statistics and Probability for PRS #242554
Just from $13/Page Let us first determine the marginal distribution of random variables U and V. For convenience, let “c=m/2” and “d=n/2”. To compute the distribution of “F”, we need to find the joint distribution of “U” and “V”. Note that, “f_U(u)=(1/(\varGamma(c)*2^c))*u^{c-1}e^{-u/2},\space u\gt0” and “f_V(v)=(1/(\varGamma(d)*2^d))*v^{d-1}e^{-v/2},\space v\gt0” Since “U” and “V” are independent, their joint distribution is given as, “f_{U,V}(u,v)=(1/\varGamma(c)\varGamma(d)2^{c+d})*u^{c-1}v^{d-1}e^{-u/2}e^{-v/2},\space u,v\gt0” We will first find the distribution of the random variable “U/V” by using the “cdf” method given as, “F_{U/V}(f)=P(U/V\leqslant f)=P(U\leqslant fV).” Now, “P(U\leqslant fV)=(1/(\varGamma(c)\varGamma(d)2^{c+d}))*\int^\infin_0(\int^{fv}_0u^{c-1}e^{-u/2}du)v^{d-1}e^{-v/2}dv” “f_{U/V}(f)= F’_{U/V}(f)=(f^{c-1}/\varGamma(c)\varGamma(d))\int^\infin_0u^{c+d-1}e^{-((f+1)/2)v}dv” Observe that, “f(v)=(f+1)^{c+d}u^{c+d-1}e^{-((f+1)/2)v}” is the “pdf”of a Gamma random variable with parameters, “\alpha=(c+d)” and “\beta=(f+1)/2”. Since “\int^\infin_0f(v)=1,” “f_{U/V}(f)=(\varGamma(c+d)/\varGamma(c)\varGamma(d))*(f^{c-1}/(f+1)^{c+d})” We now have that, “f_F(f)=f_{(U/m,V/n)}(f)=f_{(n/m)(U/V)}(f)=(m/n)f_{U/V}(m/n*f)” “=\varGamma((m+n)/2)m^{m/2}n^{n/2}f^{(m/2)-1}/\varGamma(m/2)\varGamma(n/2)(n+mf)^{(m+n)/2},\space f\gt0” Which is an “F” distribution with “m” numerator degrees of freedom and “n” denominator degrees of freedom. What Will You Get? We provide professional writing services to help you score straight A’s by submitting custom written assignments that mirror your guidelines. Premium Quality Get result-oriented writing and never worry about grades anymore. We follow the highest quality standards to make sure that you get perfect assignments. Experienced Writers Our writers have experience in dealing with papers of every educational level. You can surely rely on the expertise of our qualified professionals. On-Time Delivery Your deadline is our threshold for success and we take it very seriously. We make sure you receive your papers before your predefined time. 24/7 Customer Support Someone from our customer support team is always here to respond to your questions. So, hit us up if you have got any ambiguity or concern. Complete Confidentiality Sit back and relax while we help you out with writing your papers. We have an ultimate policy for keeping your personal and order-related details a secret. Authentic Sources We assure you that your document will be thoroughly checked for plagiarism and grammatical errors as we use highly authentic and licit sources. Moneyback Guarantee Still reluctant about placing an order? Our 100% Moneyback Guarantee backs you up on rare occasions where you aren’t satisfied with the writing. Order Tracking You don’t have to wait for an update for hours; you can track the progress of your order any time you want. We share the status after each step. Areas of Expertise Although you can leverage our expertise for any writing task, we have a knack for creating flawless papers for the following document types. Trusted Partner of 9650+ Students for Writing From brainstorming your paper's outline to perfecting its grammar, we perform every step carefully to make your paper worthy of A grade. Preferred Writer Hire your preferred writer anytime. Simply specify if you want your preferred expert to write your paper and we’ll make that happen. Grammar Check Report Get an elaborate and authentic grammar check report with your work to have the grammar goodness sealed in your document. One Page Summary You can purchase this feature if you want our writers to sum up your paper in the form of a concise and well-articulated summary. Plagiarism Report You don’t have to worry about plagiarism anymore. Get a plagiarism report to certify the uniqueness of your work. Free Features$66FREE

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