1. Show thatY=1/[1+(r_1/r_2)W], where has an F –distribution with parameters r_1 and r_2,has a beta distribution.2. LetX_1,X_2 be iid with common distribution having the pdf f(x)= e^-x, 0<∞<,zero elsewhere. Show that Z=X_1/X_2 has an F- distribution.<<>3. LetX_1,X_2 and X_3 be three independent chi-square variables with r_1, r_2, and r_3 degrees of freedom, respectively. (a) Show that Y_1=X_1/X_2 and Y_2=X_1+X_2 are independent and that Y_2isχ^2 (r_1+r_2). (b) Deduce that (X_1/r_1)/(X_2/r_2) and(X_3/r_3)/[(X_1+X_2)/(r_1+r_2)] are independent F-variables.4. Let X_1 and X_2 be two independent randomvariables so that the variances of X_1 and X_2 are σ_1^2 = k and σ_2^2, respectively. Given that thevariance of Y=3X_2-X_1 is 25, find k5. Let X and Y be independentrandom variables with means μ_1,μ_2 andvariances σ_1^2,σ_2^2. Determine the correlation coefficientof X and Z = X-Y in terms ofμ_1,μ_2, σ_1^2,σ_2^2.6. A person rolls a die, tosses acoin, and draws a card from an ordinary deck. He receives $3 for each point upon the die, $10 for a head and $0 for a tail, and $1 for each spot on the card( jack=11, queen=12, king=13). If we assume that the three random variablesinvolved are independent and uniformly distributed, compute the mean andvariances of the amount to be received.7. Let X_1, X_2 and X_3 be randomvariables with equal variances but with correlation coefficients ρ_12 =0.3, ρ_13 =0.5, ρ_23 =0.2. Find the correlationcoefficient of the linear functions Y=X_1+X_2 and Z=X_2+X_3.8. Let X be N(μ,σ^2) and consider the transformation X=log(Y) or, equivalently, Y =e^x. (a) Findthe mean and the variance of Y by first determining E(e^x) and E[(e^x)^2], byusing the mgf ofX. (b) Findthe pdf of Y. This is the pdf of the lognormal distribution.
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