3.5. Prove that the renewal function m(t), 0 uniquely determines the interarrival distribution F
Question 1: Let T be a continuous random variable with survivor function S(t). The mean residual life function m(t) is defined as m(t) = E(T-t|T > t). a. Prove that m(t) =「S(x)dz Obtain S(t) in terms of m(t), showing that m(t) uniquely define the distribution of T S(t) b.
The waiting time T between successive occurrences of an event E in a discrete-time renewal process has the probability distribution P(T- 2)0.5 and P(T 3)-0.5. a) Find the generating function U(s) for this process and hence or otherwise find the [4 probabilities u, us and e (b) The waiting time to the fifth renewal is denoted by W (i) Find the range of Ws (ii) Find the probability P(Ws- 13).
The waiting time T between successive occurrences of an event...
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R.
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
5. Let f(t) be the probability density function, and F(t) be the corresponding cumulative f(t) distribution function. Define the hazard function h(t) Show that if X is an 1-F(t): exponential random variable with parameter 1 > 0, then its hazard function will be a constant h(t) = 1 for all t > 0. Think of how this relates to the memorylessness property of exponential random variables.
Problem 2 If the cumulative distribution function of X is given by o F(b) = b<0 0<b<1 1<b<2 2<b<3 3<b<3.5 b> 3.5 1 calculate the probability mass function of X.
How to prove this? Laplace transform
If f(t) satisfies those conditions f(t) is even function which has the property of f(t) = f(-t) Laplace transform of f(t) is F(s) then Laplace transform of f(at) is F(s/lal)/lal for any non-zero constant a Prove this - How about odd function f(t) which has the property of f(-t) -f(t) ?
If f(t) satisfies those conditions f(t) is even function which has the property of f(t) = f(-t) Laplace transform of f(t) is F(s)...
Applied Mathematics Laplace Transforms
1. Consider a smooth function f(t) defined on 0 t<o, with Laplace transform F(s) (a) Prove the First Shift Theorem, which states that Lfeatf(t)) = F(s-a), where a is a constant. Use the First Shift Theorem to find the inverse trans- form of s2 -6s 12 6 marks (b) Prove the Second Shift Theorem, which states that L{f(t-a)H(t-a))-e-as F(s), where H is the Heaviside step function and a is a positive constant. Use the First and...
A point moves along a straight path. The function f(t) = log (t) determines the distance (in meters) the point has traveled in terms of the number of seconds t since the point started moving. a. How far has the point traveled 15 seconds after it started moving? meters Preview b. If the point has traveled 1.40368 meters, how many seconds have elapsed since it started moving? seconds Preview c. Write a function f that determines the number of seconds...
Suppose that f is twice differentiable function where
f(0)=f(1)=0.
Prove that
strategy Suppose that f is a twice differentiable function where f(0) = f(1) = 0. 1 Prove that f f"(x)f (x) dx a. Using part a, show that if f"(x) = wf (x) for some constant w, then w 0. Can you think of a function that satisfies these conditions for some nonzero w? b.
strategy Suppose that f is a twice differentiable function where f(0) = f(1) =...
1. Suppose t hat Xhas t he chi-square distribution on p1∈(0, ∞) degrees of f reedom and that, i ndependently, Y has t he chi-square distribution on p2∈(0, p1) degrees of f ree-dom. a. Use moment generating functions to find the distribution of X + Y . b. A naive guess might be that the distribution of X − Y is chi-square on p1− p2 degrees of freedom. Prove that such a guess is wrong by demonstrating that P (X...