Random variable X and Y have the joint PDF fXY(x,y)={c, x≥0,y≥0,(x2+y)≤1,0, otherwise Find the marginal PDF fX(x).

The marginal PDF fX(x) is obtained as given below: fX,Y(x,y)={32; x≥0,y≥0,(x2+y)≤10; otherwise} f(x)=∫yf(x,y)dy =∫01−x232dy =[32y]01−x2 =32(1−x2) f(x)=32(1−x2);0≤x≤1

Expert Answer to "Random variable X and Y have the joint..."

College
Statistics and Probability
College Statistics
Random variables

Tahmid Knox

Answered question

2021-10-25

Random variable X and Y have the joint PDF
$f_{X Y} (x, y) = {\begin{cases} c, x \geq 0, y \geq 0, (x^{2} + y) \leq 1, \\ 0, o t h e r w i s e \end{cases}$
Find the marginal PDF $f_{X} (x)$ .

Answer & Explanation

ensojadasH

Skilled2021-10-26Added 100 answers

The marginal PDF $f_{X} (x)$ is obtained as given below:
$f_{X, Y} (x, y) = {\begin{matrix} \frac{3}{2}; x \geq 0, y \geq 0, (x^{2} + y) \leq 1 \\ 0; o t h e r w i s e \end{matrix}}$
$f (x) = \int_{y} f (x, y) d y$
$= \int_{0}^{1 - x^{2}} \frac{3}{2} d y$
$= {[\frac{3}{2} y]}_{0}^{1 - x^{2}}$
$= \frac{3}{2} (1 - x^{2})$
$f (x) = \frac{3}{2} (1 - x^{2}); 0 \leq x \leq 1$

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