15 The kinetic energy k associated with a mass m moving at velocity v is given by the expression k= f(v) = mv² 2 Consider a device that fires a serrated nail into concrete at a mean velocity of 2000 feet per second, where the random velocity V possesses a density function given by v³ e-v/500 (500)4T (4) V≥ 0.
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Find the expected kinetic energy and the variance kinetic energy associated with a nail of mass m.
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- A college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X = the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is f(x)= {kx2O < x < 2} 0 otherwise a. Find the value of k and draw the corresponding density curve.[ Hint: Total area under the graph of f (x) is 1.] b. What is the probability that the lecture ends within 1min of the end of the hour? c. What is the probability that the lecture continues beyond the hour for between 60 and 90 sec? d. What is the probability that the lecture continues for at least 90 sec beyond the end of the hour?The time between successive clicks on an ad has a density function fX(x) = 2e^(−2x), x ≥ 0 and f = 0 otherwise. Find the standard deviation σX.A continuous random variable X that can assume values between x=2 and x=4 has a density function given by f(x)=12. (a) Show that the area under the curve is equal to 1. (b) Find P(3<X<3.3). (c) Find P(X≤3.9).
- A bottling company produces bottles of fruit juice that are filled, onaverage, with 32 ounces of juice. Due to random fluctuations in the machinery, the actual volume of juice is normally distributed with a standarddeviation of 0.4 oz. Let P be the probability of a bottle having less than31 oz. Express P as an integral of an appropriate density function andcompute its value numerically.A variable has the density curve whose equation is y = 1 for 0< x< 1, and y = 0 otherwise. a. Graph the density curve of this variable.b. Show that the area under this density curve to the left of any number x between 0 and 1 equals x. What percentage of all possible observations of the variablec. lie between 1/4 and 3/4?d. exceed 7/8?Suppose the amount of time (x) to complete an exam is uniform between 30 and 60 minutes. A) Draw the density curve for for the unknown height (h) B) What is the probability that the exam takes between 30 and 35 minutes or 55 and 60 minutes? Enter your answer as a fraction.
- A random variable is called a power law random variable ifP(X > t) = max{1/(t^α), 1}for some α. What is the density of X? Compute the mean of X when α > 1. What happens if α < 1?The lifetimes X and Y of two components in a machine have the joint density function f(x, y) = (1/4)*(2y + 2 − x) for 0 < x < 2, 0 < y < 1 and f(x, y) = 0 otherwise. (a) What is the probability density of the time until neither of two components is still working? (b) What is the probability distribution of the amount of time that the lifetime X survives the lifetime Y? KINDLY REQUEST YOU TO PROVIDE ME WITH COMPLETE SOLUTIONWhile taking a walk along the road where you live, you accidentally drop your glove, but you don't know where. The probability density p(x) for having dropped the glove x kilometers from home (along the road) is p(x)=7e−7x for x≥0 At what distance y from home is the probability that you dropped it within y km of home equal to 0.95?
- Let X have a uniform distribution on the interval 0 to with density function for 0 <= x <= theta; f(x)= aligned 1/ theta,\\ 0,0. aligned otherwise. (a) Find the survival function of X. (b) Find the hazard rate of X. (c) Find the mean residual-life function.A student has 3 volumes of short stories and 2 novels on a bookshelf. She selects 3 books at random to take home over vacation. A random variable X is defined to be the number of novels selected. Find the values assumed by X and its density function.. 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.