g 5. (Sec. 5.1) Suppose the marginal pdfs of two independent rvs X and Y are given byfX(x) = (5x4 for 0  x  10 otherwisefY (y) = (2y3 + y for 0  y  10 otherwise(a) Find the joint pdf of X and Y .

f(x,y) = 5x⁴(2y³ +y )           0<x<1 , 0<y<1 , 0 otherwise

Step-by-step explanation:

Given

F(x) = 5x⁴ for 0<x<1 and 0 otherwise

f(y)= 2y³ +y for 0<y<1 and 0 otherwise

the joint p.d.f is given by

f(x,y) = f(x) * f(y)= 5x⁴(2y³ +y ) 0<x<1 , 0<y<1 , 0 otherwise

The joint pdf is found by multiplying the marginal pdfs.

The two random variables are said to be statistically independent if the joint probability function can be expressed as the product of two marginal functions.

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

the correct option is b

step-by-step explanation: