Assume that X and Y are jointly continuous random variables with joint probability density function given by f(x,y)={136(3x−xy+4y) if 0&lt;x&lt;2 and 1&lt;y&lt;30 othrewise Find the marginal density functions for X and Y .

Step 1 The joint density function for X and Y is given below: f(x,y)={136(3x−xy+4y) if 0&lt;x&lt;2 and 1&lt;y&lt;30 othrewise The marginal density function of X, g(x) is obtained as given below: g(x)=∫13136(3x−xy+4y)dy =136(3xy−xy22+4y22)13 =136(2x+16) =236(x+8) =118(x+8) ={118(x+8), for 0&lt;x&lt;20, otherwise The marginal density function of y, h(y) is obtained as given below: h(y)=∫02136(3x−xy+4y)dx =136(3x22−x22y+4xy)02 =636(1+y) =16(y+1) ={16(y+1), for 1&lt;y&lt;30, otherwise

"Assume that X and Y are jointly continuous..." was answered by our experts

Bevan Mcdonald

Answered question

2021-06-03

Assume that X and Y are jointly continuous random variables with joint probability density function given by

f (x, y) = {\begin{cases} \frac{1}{36} (3 x - x y + 4 y) i f 0 < x < 2 a n d 1 < y < 3 \\ 0 o t h r e w i s e \end{cases}

Find the marginal density functions for X and Y .

Answer & Explanation

SkladanH

Skilled2021-06-04Added 80 answers

Step 1
The joint density function for X and Y is given below:

f (x, y) = {\begin{cases} \frac{1}{36} (3 x - x y + 4 y) i f 0 < x < 2 a n d 1 < y < 3 \\ 0 o t h r e w i s e \end{cases}

The marginal density function of X, g(x) is obtained as given below:

g (x) = \int_{1}^{3} \frac{1}{36} (3 x - x y + 4 y) d y

= \frac{1}{36} (3 x y - x \frac{y^{2}}{2} + 4 \frac{y^{2}}{2})_{1}^{3}

= \frac{1}{36} (2 x + 16)

= \frac{2}{36} (x + 8)

= \frac{1}{18} (x + 8)

= {\begin{cases} \frac{1}{18} (x + 8), f o r 0 < x < 2 \\ 0, o t h e r w i s e \end{cases}

The marginal density function of y, h(y) is obtained as given below:

h (y) = \int_{0}^{2} \frac{1}{36} (3 x - x y + 4 y) d x

= \frac{1}{36} (3 \frac{x^{2}}{2} - \frac{x^{2}}{2} y + 4 x y)_{0}^{2}

= \frac{6}{36} (1 + y)

= \frac{1}{6} (y + 1)

= {\begin{cases} \frac{1}{6} (y + 1), f o r 1 < y < 3 \\ 0, o t h e r w i s e \end{cases}

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