Show that the MTTF of a time to a failure distribution random variable of T can be also expressed in the following form.
Hint: The relationship between pdf, CDF or 1- R(t), and basic integration by part knowledge is needed to show the above equation holds.
Show that the MTTF of a time to a failure distribution random variable of T can...
Suppose that U is a random variable with a uniform distribution on (0,1). Now suppose that f is the PDF of some continuous random variable of interest, that F is the corresponding CDF, and assume that F is invertible (so that the function F-1 exists and gives a unique value). Show that the random variable X = F-1(U) has PDF f(x)—that is, that X has the desired PDF. Hint: use results on transformations of random variables. This cute result allows...
Problem 5. Suppose that the continuous random variable X has the distribution fx(x), -00 <oo, which is symmetric about the value r 0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number. Hint: Use integration by parts
Problem 3: The length of time to failure (in hundreds of hours) for a transistor is a random variable X with the CDF given below: 2 F(x)lTe; x20 (a) Plot the CDF by hand. (b) Derive the pdf of this random variable. (c) Compute the P(Xs0.4) 0; x<0 (d) Compute the probability that a randomly selected transistor operates for at least 200 hours. Problem 3: The length of time to failure (in hundreds of hours) for a transistor is a...
Assume the continuous random variable X follows the uniform [0,1] distribution, and define another random variable We were unable to transcribe this imagea) Determine the CDF of Y. Hint: start by writing P(Y ), then show that P(Y y) = P(X s g(v)), where g(y) is a function that you need to determine. b) Determine the PDF of Y.
1. Find Fx in terms of φ (t). Is X a continuous random variable ? 2. Compute p(X 0) 3. Compute E(X). Hint: use the CDF expectation formula, and integration by parts. You may assume that lim, t"o(-t) 0 for all n 2 0. 4. Find the CDF Fx (u) 5. Compute V(X). Hint: use Fxa, and follow the same hint of part (3) 1. Find Fx in terms of φ (t). Is X a continuous random variable ? 2....
Problem 3: Assume the continuous random variable X follows the uniform[0,1] distribution, and define another random variable Y- In () 1-X a) Determine the CDF of Y. Hint: start by writing P(Y y), then show that P(Y y) = P(X s g(v)), where g(y) is a function that you need to determine. b) Determine the PDF of Y.
2. If a random variable T has failure rate function h(t) - a + bt,t > 0, find the pdf and reliability function ofT
Suppose T is a continuous random variable whose probability is determined by the ex- ponential distribution, f(t), with mean μ. a. Compute the probability that T is less than p b. The median of a continuous random variable T is defined to be the number, m, such that P(T which mIn other words, if f(t) is the PDF of T, it is the number m for P(T )f(t) dt Compute the median for the exponential random variable T above. Is...
Suppose the random variable X represents the time to failure (in thousands of miles driven) of the signal lights on an automobile, and that X has a Weibull distribution with alpha = .0125 and Beta = 2/3. A) What is the probability that the signal lights function for at least 20,000 miles? B) For how many miles can the signal lights be expected to last? s. Suppose the random variable represents the time to failure (in thousands of miles driven)...
5. (15 Points) Let T be a random variable that is the time to failure (in years) of certain type of electrical component. T has an exponential probability density function f(x,A) =e, if >0 10, otherwise. Compute the probability that a given component will fail in 5 years or less. 5. (15 Points) Let T be a random variable that is the time to failure (in years) of certain type of electrical component. T has an exponential probability density function...