4. (10 marks) The proportion of time, X, a machine is not in operation during a...
1. (15 marks) The joint density of X and Y is given as f(x.))-o, else a) (5 marks) Find the probability density function of W-Y", where a>0 b) (5 marks) Find the probability density function of U- XY 0(5 marks) Find the probability density function of V-X-Y
3. (5 marks) The joint density of X and Y is given as else Find the probability density function of U-X-Y.
f(x,y)= 0 1. (15 marks) Suppose X and Y are jointly continuous random variables with probability density function 12, 0<x<1, 0<y<0.5 else a) (5 marks) Find P(X - Y <0.25). b) (5 marks) Find P(XY <0.30). c) (5 marks) Find V (2x - 5Y+30).
6. (5 marks) The joint density of X and Y is given as 0, else Find the probability density function of U-Y/X. Use any method you wish.
6. (5 marks) The joint density of X and Y is given as 0, else Find the probability density function of U-Y/X. Use any method you wish
PHYS1047 a) Given a random variable x, with a continuous probability distribution function fx) 4 marks b) The life expectancy (in days) of a mechanical system has a probability density write down equations for the cumulative distribution C(x) and the survival distribution Px). State a relationship between them. function f(x)=1/x, for x21, and f(x)=0 for x <1. i Find the probability that the system lasts between 0 and I day.2 marks i) Find the probability that the system lasts between...
f(x,y)=0 2. (20 marks) Suppose X and Y are jointly continuous random variables with probability density function fc, 0<x<1, 0<y<1, x + y>1 else a) (2.5 marks) Find the constant, c, so that this is valid joint density function. b) (5 marks) Find P(Y > 2X). c) (5 marks) Find P(X>0.5 Y = 0.75). d) (5 marks) Find P(X>0.5 Y <0.75). e) (2.5 marks) Are X and Y independent? Justify your answer citing an appropriate theorem.
Use Multivariate Transformations 7. (10 marks) The joint density of X and Y is given as f(x,y)=10, else a) (5 marks) Find the joint probability density function of = X / y and V = X. b) (5 marks) Using your result in (a), find the marginal density of V.
Question 1(a&b)
Question 3 (a,b,c,d)
QUESTION 1 (15 MARKS) Let X and Y be continuous random variables with joint probability density function 6e.de +3,, х, у z 0 otherwise f(x, y 0 Determine whether or not X and Y are independent. (9 marks) a) b) Find P(x> Y). Show how you get the limits for X and Y (6 marks) QUESTION 3 (19 MARKS) Let f(x, x.) = 2x, , o x, sk: O a) Find k xsl and f(x,...
Use the Method of Distribution Functions 3. (5 marks) The joint density of X and Y is given as , else Use the Method of Distribution Functions Find the probability density function of U- X-Y.