Please help me with those questions; thank you (3) Consider two variables Y₁, Y₂ whose joint probability density func-tion is given byf(y1, y2)=[3y₁ if 0 ≤ y2 ≤ Y₁ ≤ 1,0otherwise.(a) Find the marginal density f₁(₁) of Y₁.(b) Compute the conditional probability P(Y2 ≥ 1/2|Y₁ = 3/4).(c) Compute E(Y2), E(Y2) and E(Y₁Y₂).(d) Compute the covariance Cov(Y₁, Y2).

(a) To find the marginal density f1(y1) of Y1, we need to integrate the joint probability density…

Answered: (3) Consider two variables Y₁, Y₂ whose…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter13: Probability And Calculus

Section13.CR: Chapter 13 Review

Problem 4CR

See similar textbooks

Similar questions

Suppose a continuous random variable X has the probability density f(x) = K (1-x²) for 0 < x < 1, and f(x) = 0 otherwise. Find (i) K (ii) Mean
4. Let X, Y be non-negative continuous random variables with probability density functions (pdf) gx(x) and gy (y), respectively. Further, let f(x, y) denote their joint pdf. We say that X and Y are independent if f(x, y) = 9x(x)hy (y) for all x, y ≥ 0. Further, we define the expectation of X to be E[X] = √rg(x)dx, to be the expectation of XY. 0 with a similar definition for Y but g replaced by h and x replaced by y. We also define E[XY] = (0,00)x (0,00) 110,00)x (0,00) 29 (x, y) dedy (0,∞) Use Fubini's theorem (which you may assume holds) to show that if X and Y are independent, then E[XY] = E[X]E[Y]. [2]
4. Let X, Y be non-negative continuous random variables with probability density functions (pdf) gx(x) and gy (y), respectively. Further, let f(x, y) denote their joint pdf. We say that X and Y are independent if f(x, y) = gx (x)hy (y) for all x, y ≥ 0. Further, we define the expectation of X to be - 1.²0⁰ with a similar definition for Y but g replaced by h and x replaced by y. We also define to be the expectation of XY. E[X] = xg(x) dx, E(XY)= (0,00)x(0,00) Tuf(x,y)dady (0,∞) (0,∞) Use Fubini's theorem (which you may assume holds) to show that if X and Y are independent, then E[XY] = E[X]E[Y].
Suppose that the random variables X and Y have a joint probability density function f(x, y) = c(x + y)² for 0≤x≤ 1 and 0 ≤ y ≤ 1. (a) Find c. (b) Let Z = (X + Y)−¹, find E[Z]. (c) Find the marginal distribution of X and Y. (d) What is Cov(X, Y)? (e) Find the probability density function of X conditional on Y = 1.5.
The random variables X and Y have a joint probability density function given by f(x, y) = way, 0 < x < 3 and 1 < y < x, and 0 otherwise.
b) A random variable Y has a probability density function defined as: -y 1 f(y)= 0 = e
The probability density of the random variable Z isgiven by f(z) = kze−z2for z > 00 for z F 0Find k and draw the graph of this probability density.
Let X and Y be continuous random variables having a joint probability density function (pdf) given by f(x, y) = e-y, (i) (ii) (iii) = 2|X = 3).
7. Let X and Y be continuous random variables with joint probability density function given by fxy(xy): Sc, (0, 0 < x < y < 1 otherwise a) Find the value of c. b) Find the marginal distributions of X and Y. c) Find the conditional distribution of Y given X. d) Find the E(Y) and Var(Y).
Suppose that the random variables X and Y have the following joint probability density function. ƒ(x, y) = ce-6x-3y, 0 < y < x. (a) Find P(X < 1, Y < (b) Find the marginal probability distribution of X.
If the random variable T is the time to failure of a commercial product and the values of its probability den-sity and distribution function at time t are f(t) and F(t), then its failure rate at time t is given by f(t)1 − F(t). Thus, thefailure rate at time t is the probability density of failure attime t given that failure does not occur prior to time t.(a) Show that if T has an exponential distribution, thefailure rate is constant. (b) Show that if T has a Weibull distribution (see Exer-cise 23), the failure rate is given by αβt β−1.
a) Let X₁, X2, X3,..., X30 be a random sample of size 30 from a population distributed with the following probability density function: f(x) = 2 0, if 0

SEE MORE QUESTIONS

Recommended textbooks for you

Calculus For The Life Sciences

Calculus

ISBN:

9780321964038

Author:

GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:

Pearson Addison Wesley,

Calculus For The Life Sciences

Calculus

ISBN:

9780321964038

Author:

GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:

Pearson Addison Wesley,

SEE MORE TEXTBOOKS