A steel cable consists of eight high-strength steel strands. The strength of each strand can be modeled by a lognormal random variable with a mean value of 50 kips and a coefficient of variation of 10%. What is the probability that the weakest strand will have a strength less than 40 kips?

Answer:

P(X< 40) = 0.014367

Step-by-step explanation:

From the given information:

Suppose X denotes the strength of each steel strand & X follows the lognormal distribution with a mean value [tex]\mu_x[/tex] = 50 kips & coefficient of variation CV = 10%

Using the formula for the coefficient of variation CV

[tex]cv = \dfrac{\sigma_x}{\mu_x}[/tex]

[tex]0.10 = \dfrac{\sigma_x}{50}[/tex]

[tex]\sigma_x = 50 (0.10)[/tex]

[tex]\sigma_x = 5[/tex]

By applying the mean and mean & variance from the lognormal distribution, we have:

[tex]\dfrac{\sigma_x^2}{\mu_x}= \dfrac{\{exp(2 \mu + \sigma^2)\} (exp(\sigma^2))-1}{exp( \mu + \dfrac{\sigma^2}{2} \}}[/tex]

[tex]\dfrac{5^2}{50}= \dfrac{\{exp(2( \mu + \dfrac{\sigma^2}{2})) \} (exp(\sigma^2))-1}{exp( \mu + \dfrac{\sigma^2}{2} \}}[/tex]

[tex]\dfrac{5^2}{50}= exp \{ 2 ( \mu + \dfrac{\sigma^2}{2}) - ( \mu + \dfrac{\sigma^2}{2}) \} \{ ( exp ( \sigma^2))-1 \}[/tex]

[tex]\dfrac{5^2}{50}=exp \{(\mu + \dfrac{\sigma^2}{2}) \} {(exp (\sigma^2))-1 \}[/tex]

[tex]\dfrac{5^2}{50}=50 \{ (exp (\sigma^2 )) -1 \}[/tex]

[tex]\dfrac{25}{50 \times 50 }= ( exp (\sigma^2 )) -1[/tex]

[tex]exp( \sigma^2) = \dfrac{25}{50 \times 50}+ 1[/tex]

[tex]exp( \sigma^2) = (\dfrac{25+2500}{2500 })[/tex]

[tex]exp( \sigma^2) =1.01[/tex]

[tex]\sigma^2 = In (1.01)[/tex]

[tex]\mathbf {\sigma^2 = 0.00995}[/tex]

However, the next process is to replace the value of [tex]\sigma^2[/tex] into the logman distributions mean.

[tex]\mu_x = exp ( \mu + \dfrac{\sigma^2}{2})[/tex]

[tex]50 = exp ( \mu + \dfrac{\sigma^2}{2})[/tex]

[tex]50 = exp ( \mu + \dfrac{0.00995}{2})[/tex]

[tex]In(50) = ( \mu + \dfrac{0.00995}{2})[/tex]

3.912023 = μ +0.004975

μ = 3.907048

Now, we know our mean to be  3.907048 and the variance to be 0.00995.

Therefore, the probability that the weakest strand will have a strength of fewer than 40 kips can be computed as follows:

P(X< 40) = P(In(x) < In(40))

[tex]P(X<40) = P(\dfrac{In(x) - \mu}{\sigma} < \dfrac{In(40) - \mu}{\sigma})[/tex]

since;

[tex]\sigma^2 = 0.00995 \\ \\ \sigma = \sqrt{0.00995} \\ \\ \sigma = 0.0998[/tex]

[tex]P(X<40) = P(Z < \dfrac{In(40) - 3.907048}{0.0998})[/tex]

P(X< 40) = P(Z < -2.18712)

Using the Excel formula =(=NORMDIST (-2.18712)

P(X< 40) = 0.014367

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