Question

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​

Answer 1

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