The time to failure T of a component is assumed to be uniformly distributed over (a,...
2.27 The time to failure T of a component is assumed to be uniformly distributed over (a, b. The probability density is thus (1)for a<isb b-a Derive the corresponding survivor function R(t) and failure rate function z(). Draw a sketch of z()
The time to failure T of a component has probability density f (
t ) as shown
(b) Derive the corresponding survivor function R ( t ) .
(c) Derive the corresponding failure rate function z ( t ) , and
make a sketch of z(t)
Note: The f(t) is a valid pdf (so we can obtain c or the height
of the triangle). Information are enough to solve this problem.
f(t) a -b a b Time t Fig. 2.27...
1. The time to failure X of a component is uniformly distributed on [0, al. Show that the MGF of X, My(t) = E[etX], is My(t) = 607?. Use this to find E[X] and Var(X).
5. Let X be uniformly distributed in [0, 1]. Given X = x, the r.v. Y is uniformly distributed in 0, x for 0<x<1 (a) Specify the joint pdf fxy(x,y) and sketch its region of support Ω XY. (b) Determine fxly(x1025). (c) Determine the probability P(X〈2Y). (d) Determine the probability P(X +Y 1)
Question #2 The probability density function of the time to failure of a component is described by the following equation where k is constant (t) kt,t is in year) elsewhere (a) Find the k value (b) Find MTTF (c) Find the failure rate function (d) Graph the failure rate function and draw a conclusion
Need help with question 2 (not question
1)
1. Suppose that (X,Y) is uniformly distributed over the region {(x, y): 0 < \y< x < 1}. Find: a) the joint density of (X, Y); b) the marginal densities fx(x) and fy(y). c) Are X and Y independent? d) Find E(X) and E(Y). 2. Repeat Exercise 1 for (X,Y) with uniform distribution over {(x, y): 0 < \x]+\y< 1}.
Problem 1. (12 Points) A point is uniformly distributed within the disk of radius 1. That is, its density is (a) find the probability that its distance from the origin is less than k, 0 Sk1 (b) determine P(x<Y). (c) determine P(X +Y < 0.5)
1. The failure rate function of an item is z ( t ) = t^-1/2. Derive: The mean time to failure. 2. A component with time to failure T has failure rate function z ( t ) = kt fort > 0 and k > 0. Determine the probability that a component which is functioning after 200 hours is still functioning after 400 hours, when k = 2.*10^-6 (hours).
Let X be a continuous random variable uniformly distributed on the unit interval (0, 1), .e X has a density f(x) = { 1, 0<r<1 f (x)- 0, elsewhere μ+ơX, where-oo < μ < 00, σ > 0 (a) Find the density of Y (b) Find E(Y) and V(Y)
It is assumed that the time customers spend in a record store is uniformly distributed between 3 and 12 minutes. Based on this information, what is the probability that a customer will spend more than 9 minutes in the record store? A) 0.33 B) 0.1111 C) 0.25 D) 0.67 The price of a bond is uniformly distributed between $80 and $85. What is the probability that the bond price will be at least $83?