calculus help please , three parts A traffic light on campus remains red for 55 seconds at a time. A car arrives at that light and finds it red. Assume that the waiting time, t seconds, at the light follows a uniform density function f.(a) Write the density function.f(t) =for Osts 55(b) Calculate the probability of waiting no more than 20 seconds at the red light. (Round your answer to 4 decimal places.)(c) What percentage of cars will wait at least 35 seconds at the red light? (Round your answer to two decimal places.)%(d) What proportion of the wait times are less than 35 seconds? (Round your answer to four decimal places.)(e) What is the average expected wait time?sec(1) What is the standard deviation of the wait times? Round to 3 decimal places.sec

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(a) Show that f(x) = 3x2 for 0 < x < 1 represents a density function.(b) Sketch a graph of this function, and indicate the areaassociated with the probability that 0.1 < x < 0.5.(c) Calculate the probability that 0.1 < x < 0.5.
Find the distribution function and density function of Y = sinX, where X is distributeduniformly between 0 and 2π.
. The thickness of a wooden shelf in mm, has probability density function f(x) = {0.75 – 0.75 (x – 5)2} : for 4</= x </ =6 and 0 otherwise Show the table of the points for x and y that are to be plotted on the graph of the function above.
In 1950 an experiment was done observing the time gaps between successive cars on the Arroyo Seco Freeway. The data show that the density function of these time gaps was given approximately by p(t)=ae−0.119t where t is the time in seconds and a is a constant.Find the constant a.
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The life (in years) of a laptop battery has a probability density function defined by P(x)=12e−x/2 for x in [0,∞). Find the probability that a randomly selected laptop battery will last between 2 and 5 years?
a. Find the 50-th percentile of X. That is to say the value of x such that P (X ≤ x) = 0.5. b. Now say you have two independent jet engines. What is the probability that only one of themwill last more than 12 months before needing to be rebuilt? c. Find the probability density function f(x) by taking the derivative of F(x) with respect to x.
In a study of intelligence, the time (in seconds) for a laboratory animal to reach a reward in a maze was found to have a probability density function f(t) = 8 t 2 , t ≥ 8 where 8 seconds is the minimum time to traverse the maze. (a) Find the probability that an animal chosen at random takes between 28 and 56 seconds. (b) Find the probability that an animal chosen at random takes more than 28 seconds given that it took less than 56 seconds.
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