2. If a random variable T has failure rate function h(t) - a + bt,t >...
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
*Suppose a device has a constant failure rate of r(t)-A, the PDF of its lifetime follows an exponential 1. determine the reliability function, R(t) 2. determine the device's mean-time-to-fail (MTTF) *Suppose a device has a constant failure rate of r(t)-A, the PDF of its lifetime follows an exponential 1. determine the reliability function, R(t) 2. determine the device's mean-time-to-fail (MTTF)
8.2-24. A random process X(t) is applied to a network with impulse response h(t) = u(t)texp(-bt) where b > 0 is a constant. The cross-correlation of X(t) with the output Y() is known to have the same form: Rxy(t) = u(t)t exp(-bt) (a) Find the autocorrelation of y(t). (b) What is the average power in Y(0)?
4.8. Let Z be a random variable with the geometric probability mass function where 0 < π < 1. (a) Show that Z has a constant failure rate in the sense that PriZ kZk1 T for k 0, 1,.... (b) Suppose Z' is a discrete random variable whose possible values are 0, 1, and for which Pr(Z'=KZ2k} = 1-π for k 0,1,.... Show that the probability mass function for Z' is p(k).
Problem #2: Suppose that a random variable X has the following probability density function. SC(16 - x?) 0<x< 4 f(x) = 3 otherwise Find the expected value of x.
(1) Suppose the pdf of a random variable X is 0, otherwise. (a) Find P(2 < X < 3). (b) Find P(X < 1). (e) Find t such that P(X <t) = (d) After the value of X has been observed, let y be the integer closest to X. Find the PMF of the random variable y U (2) Suppose for constants n E R and c > 0, we have the function cr" ifa > 1 0, otherwise (a)...
Problem 3. The random variable X has density function f given by 0, elsewhere (a) Assuming that 6 0.8, determine K (b) Find Fx(t), the c.d.f. of X (c) Calculate P(0.4 <X < 0.8)
I. Find E(T) for a component that reliability function R(t) = pe-At, t > 0,0 < p 1 and λ > 0
2.5.9. The random variable X has a cumulative distribution function for xo , for xsO . for r>0 F(x) = z? 1 +x2 Find the probability density function of X.
2.6.17. The probability density function of the random variable X is given by 6x-21-3 -, 2<x<3 0, otherwise. Find the expected value of the random variable X.