Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0
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![Suppose that \( X \) and \( Y \) are continuous random variables with joint pdf given by
\[
f(x) =
\begin{cases}
c(x^2 + y^2) & 0 < x < 1, \, 0 < y < 1 \\
0 & \text{elsewhere}
\end{cases}
\]
1. Find \( c \).
2. Find \( f_Y(y) \), the marginal density of \( Y \).
3. Find \( E(Y) \).
4. Find the conditional density of \( X \) given \( Y = y \).
5. Find \( P(X \leq \frac{1}{2} \mid Y = 1) \).
6. Are \( X \) and \( Y \) independent? Why or why not?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcbb74936-c73d-423e-b5c0-4a7ff906acfc%2Fe3d5d21e-266e-4abd-a46b-b5fc2cce9847%2Foqdde2v_processed.jpeg&w=3840&q=75)
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- For x1, the marginal distribution fx1 (x1) function is validis it a density function)(3) Find the density of Y = X² when X ~ Binomial(4, ). The following answers are proposed: (a) P(Y = k) = ()/2*, k = 0,1, 2, 3, 4. (b) P(Y = k) = (()/2* )", k= 0,1,2,3, 4. (9) %3D %3D %3D (c) P(Y = /F) = k = 0,1, 2, 3, 4. 24 (8) (d) P(Y = k²) = k = 0, 1, 2, 3, 4. 24 (e) None of the above9. Suppose (X, Y) have the joint density f(x, y) = cxy, x, y ≥ 0; x+y≤ 1. • (a) Find the normalizing constant c. • (b) Are X, Y independent? • (c) Find the marginal densities and expectations of X, Y. • (d) Find the conditional expectation of Y given X = x.
- (i) Continuous RVs X and Y have a joint PDF (m+n+2)! S(x, y) = (1-x)"y". m!n! for 0 < yThe continuous random variables X and Y are statistically independent and have marginal density functions fx(x) = 2x, 0 1. y2' Calculate the probability P(X < 0.5, Y < 2)Consider two continuous random variables X and Y with marginal distributions g(x) and h(y)respectively and the joint density function given by: x > 0, 0 < y < 2 elsewhere. f (r.y) Then: X and Y ar statistically dependent f(x,y)#g(x)h(y) None of these f(ylx)=g(x)33.-1) Consider a continuous RV, Y, with a pdf given by f(y)=ce²¹, -∞1. The joint density for the random variables (X, Y) is given below. for 0SS1,0SySI 0. for clsewhere (a) Find the marginal density of X (b) Find the marginal density of Y (c) Find VarlX] (d) Find VarV] (e) Are X and Y statistically independent? () Find P(X>0.6) (R) I EXY-0.8, find the correlation between X and YRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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