Question

EX: Let X₁ and X₂ have the ķ $ joint proib. density function. f(x₁, x₂ ) = [ Kx1,x2, 0<=x1,<=1,0<=x2<=1 and 0 elsewhere.find the value of k

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Answer 1

17

Explanation:

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Answer 2

17

Explanation:

To find k, we use

∫∞−∞∫∞−∞fXY(x,y)dxdy=1.

Thus, we have

1=∫∞−∞∫∞−∞fXY(x,y)dxdy=∫10∫10x+cy2dxdy=∫10[12x2+cy2x]x=1x=0dy=∫1012+cy2dy=[12y+13cy3]y=1y=0=12+13c.

Therefore, we obtain c=32.

To find P(0≤X≤12,0≤Y≤12), we can write

P((X,Y)∈A)=∬AfXY(x,y)dxdy,for A=0≤x,y≤1.

Thus,

P(0≤X≤12,0≤Y≤12)=∫120∫120(x+32y2)dxdy=∫120[12x2+32y2x]120dy=∫120(18+34y2)dy=332.

We can find marginal PDFs of X and Y from their joint PDF. This is exactly analogous to what we saw in the discrete case. In particular, by integrating over all y's, we obtain fX(x). We have

Marginal PDFs

fX(x)=∫∞−∞fXY(x,y)dy, for all x,fY(y)=∫∞−∞fXY(x,y)dx, for all y.

Example

In Example 5.15 find the marginal PDFs fX(x) and fY(y).