Answer:
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).