Evaluating Deflection Probability for Lognormal Distributions

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Additionally, since lnW, lnE, and lnI are all normally distributed, their sum (ln .0069 + lnW + 4lnL - lnE - lnI) will also be normally distributed. This allows us to use the properties of the normal distribution to calculate the probability of the deflection exceeding the code-specified limit. In summary, the conversation discusses the evaluation of the probability of a beam's deflection exceeding a code-specified limit, taking into account the lognormal distribution of variables such as load, modulus of elasticity, and moment of inertia. The solution involves transforming the lognormal distribution into a normal distribution using the natural logarithm, and then using the properties of the normal distribution to calculate the desired probability
  • #1
omgitsroy326
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I'm quite confused bout lognormal distribution and normal distribution

Consider a three-span continuous beam. All supports are pinned supports. Each span has a length of l. The beam
has a modulus of elasticity E and a moment of inertia I. All spans are subjected to a uniformly distributed load W.
The maximum deflection of the beam occurs in the outer spans and is equal to

0.0069Wl^4 / (EI)


(a) Your job is to evaluate the probability pF that the deflection will exceed the code-specified limit of l/360
assuming that W, E, and I are statistically independent lognormal random variables and given the following
information:
l = 5 m (deterministic)
W has a mean value of 10 kN/m and a coefficient of variation of 0.4.
E has a mean value of kN/m2 and a coefficient of variation of 0.25.
I has a mean value of m4 and a standard deviation of m4.

I have the solution for it but I'm lost on when

Variables W E I are all lognormal distribution.
Now to solved for the mean you would do
d = 0.0069Wl^4 / (EI)

Now since you can't divide or multiply lognormal distributions you would have to ln the entire formual of d to be:

ln d = ln .0069 +lnW + 4ln L - ln E - ln I

since you're takin the natural log of a lognormal distribution it becomes a normal distribution ... Why is this ? I'm confused on this part .
 
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  • #2
It helps to think of the lognormal distribution as a transformation of the normal distribution. The lognormal distribution is the exponential of a normal distribution, which means it is the result of multiplying (or raising to a power) a normal random variable by itself. Since the natural logarithm (ln) is the inverse of the exponential, taking the natural logarithm of a lognormal random variable will transform the lognormal random variable back into a normal random variable. This explains why ln d is a normal distribution in this case.
 
  • #3


The reason why taking the natural log of a lognormal distribution results in a normal distribution is due to the properties of logarithms. When you take the natural log of a lognormal distribution, you are essentially taking the log of the exponent of the distribution. This results in a transformation of the distribution into a normal distribution.

In other words, the natural log of a lognormal distribution removes the exponential component and converts it into a linear form, which is the characteristic of a normal distribution. This transformation allows for easier mathematical analysis and calculations.

In the context of evaluating deflection probability for lognormal distributions, taking the natural log of the formula allows us to solve for the mean and standard deviation of the normal distribution, which can then be used to calculate the probability of the deflection exceeding the code-specified limit.

Overall, understanding the properties and transformations of different distributions is important in statistical analysis and probability calculations.
 

1. What is a lognormal distribution?

A lognormal distribution is a probability distribution that is used to represent a set of data where the logarithms of the values are normally distributed. This means that the data is skewed to the right, with most of the values falling towards the lower end and a few extreme values towards the higher end.

2. What are the characteristics of a lognormal distribution?

The characteristics of a lognormal distribution include a positive skew, as mentioned before, and a mean, median, and mode that are not equal. The shape of the distribution is determined by two parameters: the location parameter, which affects the location of the peak, and the scale parameter, which affects the spread of the data.

3. What are some real-world applications of lognormal distributions?

Lognormal distributions are commonly used in fields such as finance, economics, and engineering. They can be used to model stock prices, incomes, and the size of earthquakes. They are also useful in the analysis of data that is constrained to be positive, such as the number of people in a population or the size of particles in a sample.

4. How is a lognormal distribution related to a normal distribution?

A lognormal distribution is related to a normal distribution in that the logarithm of a lognormal variable follows a normal distribution. This means that if we take the natural logarithm of the data, the resulting values will be normally distributed. However, the raw data itself will not be normally distributed.

5. How is a lognormal distribution different from a normal distribution?

One key difference between a lognormal distribution and a normal distribution is that the former is skewed to the right, while the latter is symmetric. Additionally, a lognormal distribution has a lower bound of zero, while a normal distribution can take on any value along the x-axis. The two distributions also have different equations and different interpretations of their parameters.

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