Y V1 - X2 Find the probability density for the new random variable Y 3v Abu Nawas is a waiter who...
question 4, 3 sub-questions
Final2.jpeg iat probability density function is given by f(x, y (2(x2+ y'), 0<x <2, 0cy3 Find the marginal densities of X and Y. Lip 2 Are X and Y independent? 4p c. Find probability X + Y < i , P(X+Y < 1 ). The amount of time, in hours, that a computer functions before breaking down .b] uniformly distributed on [o is continuous random variable T
2. Let R be the region R = {(X,Y)|X2 + y2 < 2} and let (X,Y) be a pair of random variables that is distributed uniformly on this region. That is fx,y(x, y) is constant in this region and 0 elsewhere. State the sample space and find the probability that the random variable x2 + y2 is less than 1, P[X2 +Y? < 1].
Question # A.4 (a) Given that probability density function (pdf of a random variable (RV), x is as follows: Px(x)-axexp(-ax) x 20 otherwise where α is a constant. Suppose y = log(x) and y is monotonic in the given range of X. Determine: (i) pdf of y; (ii) valid range of y; and, (iii) expected value of y. Answer hint:J exp(y) (b) Given that, the pdf, namely, fx(x) of a RV, x is uniformly distributed in the range (-t/2, +...
Problem 2 If Xi, X2. ,Xso be independent and idatically distributed with probability density function same as random variable X (x) = 1/2e-2x x > 0 and Y-X1 X2+X Points 5 Points) 5 Points a) Find Moment Generating Function of Y, My(S) b) What is MGF of-2x c What is MGF of 2X +3
Problem 2 If Xi, X2. ,Xso be independent and idatically distributed with probability density function same as random variable X (x) = 1/2e-2x x > 0...
Problem 5 . This question considers uniform random points on the unit disc x2+92 〈 1 (a) A point (X, Y) is uniformly chosen in the unit disc. Find the CDF and PDF of its distance from the origin R X2 +Y2 (b) Compute the expected distance from the origin. (c) Determine the marginal PDF of X and Y (d) Are X and Y independent? (Justify your claims) e) One way to generate uniform random points on this disc is...
2. (5 marks) Consider a random variable Y with density function 31 ,else Find the probability density function of U 4-Y2
Show the random variables X and Y are independent, or not
independent
Find the joint cdf given the joint pdf below
Suppose that (X, Y) is uniformly distributed over the region defined by 0 sys1-x2 and -1sx 4 Therefore, the joint probability density function is, 0; Otherwise
Suppose that (X, Y) is uniformly distributed over the region defined by 0 sys1-x2 and -1sx 4
Therefore, the joint probability density function is, 0; Otherwise
I . (20%) Random variable X has the probability density function as ; Random variable Y 2X+1 0 otherwise a) Determine A b) Determine the Probability Distribution Function F, (x) c) Determine E(X) and ơx d) Determine the probability density function fy(y) and E(Y)
3. (10 points) Let X be continuous random variable with probability density function: fx(x) = 7x2 for 1<<2 Compute the expectation and variance of X 4. (10 points) Let X be a discrete random variable uniformly distributed on the integers 1.... , n and Y on the integers 1,...,m. Where 0 < n S m are integers. Assume X and Y are independent. Compute the probability X-Y. Compute E[x-Y.
9. Let Y be a random variable who probability density function is given by P(Y- y) = 옮 when y is an intger between 1 and n inclusive. What should n be for this to be valid probability density function? Compute E(Y), V(Y), E(17Y - T) and 10, 10% of bottles produced at a factory have cracks. If two bottles are selected, find the mean and variance of the number of cracked bottles selected.