⧼exchistory⧽
23 exercise(s) shown, 0 hidden
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BBy Bot
Jun 09'24
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Suppose you choose at random a real number

[math]X[/math] from the interval [math][2,10][/math].

  • Find the density function [math]f(x)[/math] and the probability of an event [math]E[/math] for this experiment, where [math]E[/math] is a subinterval [math][a,b][/math] of [math][2,10][/math].
  • From (a), find the probability that [math]X \gt 5[/math], that [math]5 \lt X \lt 7[/math], and that [math]X^2 - 12X + 35 \gt 0[/math].
BBy Bot
Jun 09'24
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Suppose you choose a real number [math]X[/math] from the interval

[math][2,10][/math] with a density function of the form

[[math]] f(x) = Cx\ , [[/math]]

where [math]C[/math] is a constant.

  • Find [math]C[/math].
  • Find [math]P(E)[/math], where [math]E = [a,b][/math] is a subinterval of [math][2,10][/math].
  • Find [math]P(X \gt 5)[/math], [math]P(X \lt 7)[/math], and [math]P(X^2 - 12X + 35 \gt 0)[/math].
BBy Bot
Jun 09'24
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Same as Exercise, but suppose

[[math]] f(x) = \frac Cx\ . [[/math]]

BBy Bot
Jun 09'24
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Suppose you throw a dart at a circular target of radius 10 inches. Assuming that you hit the target and that the coordinates of

the outcomes are chosen at random, find the probability that the dart falls

  • within 2 inches of the center.
  • within 2 inches of the rim.
  • within the first quadrant of the target.
  • within the first quadrant and within 2 inches of the rim.
BBy Bot
Jun 09'24
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Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density

[[math]] f(t) = \lambda e^{-\lambda t}\ , [[/math]]

where [math]\lambda = 1[/math], so that the probability [math]P(0,T)[/math] that a particle will appear in the next [math]T[/math] seconds is [math]P([0,T]) = \int_0^T\lambda e^{-\lambda t}\,dt[/math]. Find the probability that a particle (not necessarily the first) will appear

  • within the next second.
  • within the next 3 seconds.
  • between 3 and 4 seconds from now.
  • after 4 seconds from now.
BBy Bot
Jun 09'24
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Assume that a new light bulb will burn out after [math]t[/math] hours, where [math]t[/math] is chosen from [math][0,\infty)[/math] with an exponential density

[[math]] f(t) = \lambda e^{-\lambda t}\ . [[/math]]

In this context, [math]\lambda[/math] is often called the failure rate of the bulb.

  • Assume that [math]\lambda = 0.01[/math], and find the probability that the bulb will not burn out before [math]T[/math] hours. This probability is often called the reliability of the bulb.
  • For what [math]T[/math] is the reliability of the bulb [math] = 1/2[/math]?
BBy Bot
Jun 09'24
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Choose a number [math]B[/math] at random from the

interval [math][0,1][/math] with uniform density. Find the probability that

  • [math]1/3 \lt B \lt 2/3[/math].
  • [math]|B - 1/2| \leq 1/4[/math].
  • [math]B \lt 1/4[/math] or [math]1 - B \lt 1/4[/math].
  • [math]3B^2 \lt B[/math].
BBy Bot
Jun 09'24
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Choose independently two numbers [math]B[/math] and [math]C[/math] at random from the interval [math][0,1][/math] with uniform density. Note that the point [math](B,C)[/math] is then chosen at random in the unit square. Find the probability that

  • [math]B + C \lt 1/2[/math].
  • [math]BC \lt 1/2[/math].
  • [math]|B - C| \lt 1/2[/math].
  • [math]\max\{B,C\} \lt 1/2[/math].
  • [math]\min\{B,C\} \lt 1/2[/math].
  • [math]B \lt 1/2[/math] and [math]1 - C \lt 1/2[/math].
  • conditions (c) and (f) both hold.
  • [math]B^2 + C^2 \leq 1/2[/math].
  • [math](B - 1/2)^2 + (C - 1/2)^2 \lt 1/4[/math].
BBy Bot
Jun 09'24
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Suppose that we have a sequence of occurrences. We assume that the time [math]X[/math] between occurrences is exponentially distributed with [math]\lambda = 1/10[/math], so on the average, there is one occurrence every 10 minutes (see Example).

You come upon this system at time 100, and wait until the next occurrence. Make a conjecture concerning how long, on the average, you will have to wait. Write a program to see if your conjecture is right.

BBy Bot
Jun 09'24
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As in Exercise, assume that we have a sequence of occurrences, but now assume that the time [math]X[/math] between occurrences is uniformly distributed between 5 and 15. As before, you come upon this system at time 100, and wait until the next occurrence. Make a conjecture concerning how long, on the average, you will have to wait. Write a program to see if your conjecture is right.