Innehåll 1 Sannolikhetsteori - Matematikcentrum

1 SANNOLIKHETSTEORI
32. [ ∗ ] The cantilever beam shown in the figure is subjected to a random concentrated load P and a
random distributed load W . Assume
P is N(5, 1), in kN
W is N(1, 0.2), in kN/m
(a) Determine the mean and variance of the applied bending moment M a = 50W+10P. Assume
thatÖW,P = 0.5 (that is, the loads are correlated).
(b) The resisting moment of the beam M r which is statistically independent of the applied moment
M a , is also Gaussian N(200, 50) in kNm. Determine the probability of failure of the
beam, P(M r < M a ) assuming that M a is Gaussian.
33. [∗] The figure shows a schematic procedure of the treatment system for the waste from a factory before
it is dumped into a nearby river. Here X denotes the concentration of a pollutant feeding into
the treatment system, and Y denotes the concentration of the same pollutant leaving the system.
Suppose that for a normal day, X has a log-normal distribution with median 4 mg/l and the coefficient
of variation (COV, variationskoefficient) is 20 %. Because of the erratic nature of biological
and chemical reactions, the efficiency of the treatment system is unpredictable. Hence the fraction
of the influent pollutant remaining untreated, denoted by F, is also a random variable. Assume F is
also a log-normal variate with a median of 0.15 and COV of 10 %. Assume X and F are statistically
independent.
(a) Determine the distribution of Y and the values of its parameters. Note that Y = F · X .
(b) Suppose that the maximum concentration of the pollutant permitted to be dumped into the
river is specified to be 1 mg/l. What is the probability that this specified standard will be
exceeded on a normal day
(c) On some working days, owing to heavy production in the factory, the influent X will have
a median of 5 mg/l instead. Assume that the distribution of X is still log-normal with the
same coefficient of variation and that the efficiency of the treatment system does not change
statistically. Suppose that such a heavy work day happens only 10 % of the time. Then, on a
given day selected at random, what is the probability that the specified standard of 1 mg/l for
Y will be exceeded
34. The maximum annual flood level of a river is denoted
by H (in meters). Assume that the probability density
of H is described by the triangular distribution shown
in the figure.
f H
(h)
(a) Determine the flood height h 20 which has a mean
recurrence interval (return period) of 20 years.
(P(H > h 20 ) = 1/20)
5 6 7
h, m
(c) What is the probability that during the next 20 years the river height H will exceed h 20 at least
once
(d) What is the probability that during the next 5 years the value of h 20 will be exceeded exactly
once
(e) What is the probability that h 20 will be exceeded at most twice during the next 5 years
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