ENGI 3423 Probability and Statistics

Faculty of Engineering and Applied Science
2010 Fall

Problem Set 6

[Continuous distributions, Normal distribution, Joint probability distributions]

To be completed by Monday 8 November, 2010.
Compare your solution with the files listed on the solution page.

  1. The concentration of pollutant in a sample taken from a lake, measured in mg/10,000l, is found to follow a normal distribution with a mean of 19.6 and a standard deviation of 4.6.   Three events are considered to be of importance:

    A = pollution in a sample being less than 15 mg/10,000l.
    B = pollution in a sample being greater than 15 but less than 25 mg/10,000l.
    C = pollution in a sample being greater than 25 mg/10,000l.

    1. What is the probability of individual events A, B, and C occurring?
    2. What is the probability that in 6 random samples, event A occurs once, B occurs 5 times and C does not occur?

  1. Two machines are available for the manufacture of resistors rated at 100 ohm, for use in a particular type of electric circuit.   A resistor is rejected if its actual resistance R is not in the range 99.90 < R < 100.10 (ohm).

    Machine ‘A’ produces resistors whose resistance is normally distributed with mean 100.00 ohm and standard deviation 0.05 ohm.
    Machine ‘B’ produces resistors whose resistance is normally distributed with mean 100.05 ohm and standard deviation 0.02 ohm.

    1. Which machine is more likely to produce an acceptable resistor?
      Show your working.

    2. The difference C of two normally distributed random quantities A, B is also normally distributed, with its mean equal to the difference of the two means:
              µC   =   µA – µB .
      If the random quantities are uncorrelated, (which is true if they are independent), then the variance of the difference is the sum of the variances:
              V[C]   =   V[A] + V[B] .

      How likely is it that a randomly chosen resistor produced by machine ‘A’ will have a lower resistance than that of a randomly chosen resistor produced by machine ‘B’?

  1. The joint probability mass function   p(x, y)   for a pair of discrete random quantities   X, Y   is defined by the following table.

    p
    y
    –101
    x –1 .10.20.20
    0 .06.08.16
    1 .04.12.04

    1. Find the correlation coefficient rho for   X   and   Y , correct to three decimal places.
    2. Are   X   and   Y   stochastically independent?

  1. An engineer obtains the following data from measurements of the 10 minute average windspeed in the east-west direction at the location of a building project.   Parts (a), (c) and (d) below may be done by hand or by a software package such as Minitab.   The data are also available in this separate text file.

    –3.06 –8.14 –0.23 5.14 4.73
    7.51 –12.85 –2.72 4.19 –7.15
    –5.09 –10.29 –11.10 –6.58 –5.52
    –12.51 –4.45 –3.60 –0.80 –3.40
    –3.20 –3.43 5.48 –1.94 –2.47
    –4.17 8.76 3.00 10.85 –4.91
    7.14 –9.88 1.30 3.19 8.48
    –1.94 –4.22 2.01 –3.48 2.44
    –9.01 –5.91 –9.41 –3.39 –1.67
    –1.35 –3.18 9.91 –10.56 –5.33
    –14.90 6.03 –8.15 –4.90 2.44
    0.93 3.05 1.60 –8.63 –7.30

    1. Construct a (relative frequency) histogram.   Use six class intervals.
    2. Use either Minitab or Excel to produce a normal probability plot for these data.   Comment on whether or not the data appear to be consistent with a normal distribution.
    3. Determine the mean, median, variance and coefficient of variation for these data.
      [The coefficient of variation is the ratio of the standard deviation to the absolute value of the mean]
    4. Is there any evidence for skewness from a boxplot of the data?

  1. [In this question, you will need the relationship that
    A ~ N(µA , sigmaA 2) ,   B ~ N(µB , sigmaB 2)   and   A, B independent
    implies A + B ~ N(µA + µB ,   sigmaA 2 + sigmaB 2) .]

    The vehicle speed distribution on a two-way road is N(80, 202), where speed is measured in km/h.

    1. Determine the probability that the relative speed of two vehicles moving in opposite directions is between 150 and 200 km/hour.

    2. Upper and lower speed limits of 100 km/h and 50 km/h are imposed on the road.   Sketch the new probability distribution of vehicle speeds assuming that the limits are obeyed and that the affected drivers change their vehicle speed by the minimum legal amount.

    3. Is the (idealized) distribution in part (b) continuous, discrete or something else?   Discuss briefly.

  1. Measurements of the power output of a motor are known to have a standard deviation of sigma = 45 W.   Find the smallest sample size needed to ensure that the sample mean of a random sample lies within 10 W of the unknown true mean   µ   at least 95% of the time.

  1. To a good approximation, the strength at failure of a valve, (expressed in terms of the applied pressure that causes failure), is known to be normally distributed with a mean of 980 kN m–2 and a standard deviation of 60 kN m–2 .   The maximum pressure to which the valve is subjected in any given week is known to be normally distributed with a mean of 725 kN m–2 and a standard deviation of 80 kN m–2.

    1. Find the probability that a randomly chosen valve fails during the first week.
    2. Find the odds that the valve does not fail during the first 52 weeks, (that is, during the first year).
      [Express the odds in the approximate form r : 1, with r rounded off to the nearest integer.]
      State any assumptions that you have made.

  1. BONUS QUESTION:

    If two independent continuous random quantities U and V have probability density functions g(u) and h(v) respectively, then their sum, X = U + V, has a p.d.f. f (x), given by the convolution

    [definition of a convolution integral]

    Show that the sum of two independent identically distributed random quantities with the p.d.f. of an exponential distribution g(x; L) = L exp(-L x) is a random quantity that has the p.d.f. f(x; 2, L) = L^2 x exp(-L x), (which is the Gamma distribution Γ(2, λ).)

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Created 2003 10 14 and most recently modified 2010 07 30 by Dr. G.H. George