Concept explainers
The special case of the gamma distribution in which α is a positive integer n is called an Erlang distribution. If we replace b by 1/λ in Expression (4.8), the Erlang
It can be shown that if the times between successive events are independent, each with an exponential distribution with parameter λ, then the total time X that elapses before all of the next n events occur has pdff (x; λ, n).
a. What is the
b. If customer interarrival time is exponentially distributed with λ = .5, what is the probability that the tenth customer (after the one who has just arrived)will arrive within the next 30 min?
c. The
Want to see the full answer?
Check out a sample textbook solutionChapter 4 Solutions
Probability and Statistics for Engineering and the Sciences
- The probability mass function of X = the number of major defects in an electrical appliance of a randomly selected type is: Calculate the following:a) E (X)b) V (X) directly from the definitionarrow_forwardSuppose the proportion X of surface area in a randomly selected quadrat that is covered by a certain plant has a standard beta distribution with ? = 4 and ? = 2. (a) Compute E(X) and V(X). (Round your answers to four decimal places.) E(X) = V(X) = (b) Compute P(X ≤ 0.3). (Round your answer to four decimal places.)(c) Compute P(0.3 ≤ X ≤ 0.7). (Round your answer to four decimal places.)arrow_forwardLet X1, . . . , Xn i.i.d. U([θ1, θ2]), i.e., X1, . . . , Xn are independent and follow a uniform distribution on the interval [θ1, θ2] for θ1, θ2 ∈ R and θ1 < θ2. Find an estimator for θ1 and θ2 using the method of moments.arrow_forward
- Suppose a random variable X has a probability density function as shown in img1.jpg c. Find P(X <= 0.4 | X <= 0.8) Note: I already solve part A and the value of k is 6.arrow_forwardIf X has the exponential distribution given by f(x) =0.5 e−0.5x, x > 0, find the probability that x > 1.arrow_forwardIf the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.arrow_forward
- Consider a random sample X1,...,Xn (n > 2) from Beta(θ,1), where we wish to estimate the parameter θ. (a) Find the MLE θˆ and write it as a function of T = − ∑ni=1 log Xi. (b) Find the sampling distribution of T = − ∑ni=1 log Xi . (Hint: First find the distribution of Ti = − log Xi .)arrow_forwardConsider a random sample X1, … , Xn from the pdff (x; u) = .5(1 + (THETA)x) -1 <= x <= 1where -1 <= theta <= 1 (this distribution arises in particlephysics). Show that theta = 3X is an unbiased estimator oftheta. [Hint: First determine mu = E(X) = E(X).]arrow_forwardLet X be an exponential random variable with standard deviation σ. FindP(|X − E(X)| > kσ ) for k = 2, 3, 4, and compare the results to the boundsfrom Chebyshev’s inequality.arrow_forward
- Glencoe Algebra 1, Student Edition, 9780079039897...AlgebraISBN:9780079039897Author:CarterPublisher:McGraw Hill