Question

14. The measured free compressive strength averages of the cylindrical ground samples are reported to be 40 MPa and the variances of 9 MPaʼ with lognormal distribution. The free compressive strength of a sample chosen randomly from these ground samples; a) Find the probability of being at most 35 MPa. b) Find the probability of being between 45 MPa and 50 MPa. c) Estimate the highest strength value of the 8% group with the lowest free pressure resistance 15. A probability distribution with the random variable X has a uniform distribution in the range (3,9). According to this Windows a) P (X = 2) = ? b) P (5 <X S 7) = ? c) P(X<5) = ? d) P (6 SX) = ? e) E(X) = ? f) Var [X] =?

Answer 1

The solution of Question 11: Average number of failure per year = 3

So, the rate parameter $\lambda = 3$

The pdf of Poisson's distribution is given by

$P(X=x) = \frac{\lambda^x.e^{-\lambda}}{x!}$

where x is the number of failures.

a) The probability that the test tool will never fail in a year is

30 e P(X =0) = = 0.05 0!

b) For a three month period, the rate parameter would be

$\lambda = \frac{3}{4} = 0.75$

The probability that the test tool will never fail in a 3-month period

$P(X=0) = \frac{0.75^0.e^{-0.75}}{0!} = 0.472$

c) For a six months period, the rate parameter would be

$\lambda = \frac{3}{2} = 1.5$

The probability that the test tool will fail twice in a 6-month period

$P(X=2) = \frac{1.5^2.e^{-1.5}}{2!} = 0.251$

d) For a 9 months period, the rate parameter would be

$\lambda = \frac{9}{4} = 2.25$

The probability that the test tool will fail 2 or more times in a 9-month period = 1 - The probability that the test tool will fail never or just once in a 9-months period

$= 1- [\frac{2.25^0.e^{-2.25}}{0!} + \frac{2.25^1.e^{-2.25}}{1!}]$

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