if any probability density of a co tinuous random variable x is f(x) =k÷1+x2,limit -infinite to + iinfinite otherwise 0 then the value in mcq
Answers
Answer:
Problem
Let X and Y be jointly continuous random variables with joint PDF
fX,Y(x,y)=⎧⎩⎨⎪⎪cx+10x,y≥0,x+y<1otherwise
Show the range of (X,Y), RXY, in the x−y plane.
Find the constant c.
Find the marginal PDFs fX(x) and fY(y).
Find P(Y<2X2).
Solution
Problem
Let X and Y be jointly continuous random variables with joint PDF
fX,Y(x,y)=⎧⎩⎨⎪⎪6e−(2x+3y)0x,y≥0otherwise
Are X and Y independent?
Find E[Y|X>2].
Find P(X>Y).
Solution
Problem
Let X be a continuous random variable with PDF
fX(x)=⎧⎩⎨⎪⎪2x00≤x≤1otherwise
We know that given X=x, the random variable Y is uniformly distributed on [−x,x].
Find the joint PDF fXY(x,y).
Find fY(y).
Find P(|Y|<X3).
Solution
Problem
Let X and Y be two jointly continuous random variables with joint PDF
fX,Y(x,y)=⎧⎩⎨⎪⎪6xy00≤x≤1,0≤y≤x−−√otherwise
Show RXY in the x−y plane.
Find fX(x) and fY(y).
Are X and Y independent?
Find the conditional PDF of X given Y=y, fX|Y(x|y).
Find E[X|Y=y], for 0≤y≤1.
Find Var(X|Y=y), for 0≤y≤1.
Solution
Problem
Consider the unit disc
D={(x,y)|x2+y2≤1}.
Suppose that we choose a point (X,Y) uniformly at random in D. That is, the joint PDF of X and Y is given by
fXY(x,y)=⎧⎩⎨⎪⎪1π0(x,y)∈Dotherwise
Let (R,Θ) be the corresponding polar coordinates as shown in Figure 5.10. The inverse transformation is given by
{X=RcosΘY=RsinΘ
where R≥0 and −π<Θ≤π. Find the joint PDF of R and Θ.
Figure 5.10: Polar Coordinates
Solution