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Using the Exponential Density Function In Exercises 33-36, find the required probabilities using the exponential probability density function
Useful Life The lifetime t (in years) of a battery is exponentially distributed with
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Calculus: An Applied Approach (MindTap Course List)
- 3.69 The life span in hours of an electrical compo- nent is a random variable with cumulative distribution function e- 50, x > 0, 10, 1 F(x) = {0. - eleswhere.arrow_forwardThe probability density function of the time a customer arrives at a terminal (in minutes after 8:00 A.M.) is f(x) = 0.25 e-x/4 for x > 0. Determine the probability that(a) The customer arrives by 10:00 A.M. (Round your answer to one decimal place (e.g. 98.7))(b) The customer arrives between 8:19 A.M. and 8:31 A.M. (Round your answer to four decimal places (e.g. 98.7654))(c) Determine the time (in hours A.M. as decimal) at which the probability of an earlier arrival is 0.51. (Round your answer to two decimal places (e.g. 98.76))(d) Determine the cumulative distribution function and use the cumulative distribution function to determine the probability that the customer arrives between 8:19 A.M. and 8:31 A.M. (Round your answer to four decimal places (e.g. 98.7654))(e) Determine the mean and (f) standard deviation of the number of minutes until the customer arrives. (Round your answers to one decimal place (e.g. 98.7))arrow_forwardPart B 1) The probability density function is f(x) = 2e-kx i) Standard deviation of x = ?.arrow_forward
- Random variable X has the following probability mass function: P(X=x) = (x+1)/12 for x = 0, 1, 2, 5. Based on the PMF, what is P(X> or equal to 2)?arrow_forward70 The probability density function of the weight of packages delivered by a post office is f(x) = for 1arrow_forwardLosses have a lognormal distribution with μ=6.9078 and σ=1.5174 Determine the percentage increase in the number of losses that exceed 1,000 if all losses are increased by 10%.arrow_forwardHAL AMI Figure 1 shows the piecewise function (I), (II) and (III) for cumulative distribution function F(x) for continuous random variable. F(x) III II 0. 2 6. Figure 1 Construct the probability density function f(x). Should one of the piecewise function (III) is not constant, explain the changes.arrow_forwardLet X be a random variable with probability density function is given byf(x) = Cx?- x/2 x>0(a) What is the value of c? (b) What is the cumulative distribution function of X?(c) F(5) = ?arrow_forwardThe arrival time t(in minutes) of a bus at a bus stop is uniformly distributed between 10:00 A.M. and 10:03 A.M. (a) Find the probability density function for the random variable t. (Let t = 0 represent 10:00 A.M.) f(t) = _______??_________ , [0,3] (b) Find the mean and standard deviation of the arrival times. (Round your standard deviation to three decimal places.) mean = _________??________________ minutes after 10:00 A.M. standard deviation = _________??_____________ min (c) What is the probability that you will miss the bus if you arrive at the bus stop at 10:02 A.M.? (Round your answer to two decimal places.) __________________??_____________________arrow_forwardTheory- Let uf and of be the expectation and variance of a forecast model F for an observations y, and let pf(:) denote the probability density function for the forecast model. Also let G be a different forecast model. Which of the following statements are true? 1. Two forecast scores S(F, y) and S(G, y) are independent 2. The squared error (y – µF)² is a proper score 3. The score (y – µf)²lof is a proper score that improves on the squared error score by taking the variance into account. 4. The logarithmic score – log[pf(y)] is a strictly proper score Enter your answer below as a vector indicating the true statements (c(1), c(1,4) or similar). R Code E Start Over Run Code M Submit Answer 1 2 Next Topicarrow_forwardOne of the important properties of the probability density function, f(x), of a random variable X is f(x) > 0, for all values of X. Select one: O True O Falsearrow_forwardMX Suppose a college professor never finishes her lecture before the end of the class period, and always finishes within five minutes after the class period is supposed to end. Let X = time that elapses between the end of the class period and the actual end of the lecture. Suppose the pdf of X is f(x) = kx³/2 0 ≤ x ≤ 5 and 0 otherwise. Find the value of k that makes f(x) a legitimate probability density function and use that value of k to find the probability that the lecture ends less than 3 minutes after the class period is supposed to end. Please write completearrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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