Question

1. (a) ( [5 ) marks] Let ( X ) be ( operatorname{Uniform}(-a, a) ) random variable, ( a>0 ). i. Determine the density function of ( Y=X^{2} ). ii. Determine the density function of ( W=sqrt{X+a} ). (b) ( [10 ) marks ( ] ) Let ( X ) and ( Y ) be independent gamma random variables with parameters ( (2, eta) ) and ( (5, eta) ), respectively. i. What is the joint density function of ( X ) and ( Y ) ? ii. Find the joint density function of ( U=X+Y ) and ( V=X /(X+ ) ( Y) ). iii. Compute the marginal density functions ( f_{U}(u) ) and ( f_{V}(v) ). iv. Are ( U ) and ( V ) independent? v. Using the moment generating functions ( m_{X}(u) ) and ( m_{Y}(u) ), determine the moment generating function ( m_{X+Y}(u) ).

          1. (a) ( [5 ) marks] Let ( X ) be ( operatorname{Uniform}(-a, a) ) random variable, ( a>0 ).
i. Determine the density function of ( Y=X^{2} ).
ii. Determine the density function of ( W=sqrt{X+a} ).
(b) ( [10 ) marks ( ] ) Let ( X ) and ( Y ) be independent gamma random variables with parameters ( (2, eta) ) and ( (5, eta) ), respectively.
i. What is the joint density function of ( X ) and ( Y ) ?
ii. Find the joint density function of ( U=X+Y ) and ( V=X /(X+ ) ( Y) ).
iii. Compute the marginal density functions ( f_{U}(u) ) and ( f_{V}(v) ).
iv. Are ( U ) and ( V ) independent?
v. Using the moment generating functions ( m_{X}(u) ) and ( m_{Y}(u) ), determine the moment generating function ( m_{X+Y}(u) ).

1. (a) ( [5 ) marks] Let ( X ) be ( operatornameUniform(-a, a) ) random variable, ( a>0 ).
i. Determine the density function of ( Y=X^2 ).
ii. Determine the density function of ( W=sqrtX+a ).
(b) ( [10 ) marks ( ] ) Let ( X ) and ( Y ) be independent gamma random variables with parameters ( (2, eta) ) and ( (5, eta) ), respectively.
i. What is the joint density function of ( X ) and ( Y ) ?
ii. Find the joint density function of ( U=X+Y ) and ( V=X /(X+ ) ( Y) ).
iii. Compute the marginal density functions ( fU(u) ) and ( fV(v) ).
iv. Are ( U ) and ( V ) independent?
v. Using the moment generating functions ( mX(u) ) and ( mY(u) ), determine the moment generating function ( mX+Y(u) ).

Added by Onder K.

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