# The flow in a river can be modeled as a log-normal distribution. From the data, it was estimated that, the probability that the flow exceeds 850 cfs is 50% and the probability that it exceeds 100 cfs is 90%. Let X denote the flow in cfs in the river. What is the standard deviation of log (to the base 10) of X? Please report your answer in 3 decimal places.

Question

The flow in a river can be modeled as a log-normal distribution. From the data, it was estimated that, the probability that the flow exceeds 850 cfs is 50% and the probability that it exceeds 100 cfs is 90%. Let X denote the flow in cfs in the river. What is the standard deviation of log (to the base 10) of X? Please report your answer in 3 decimal places.

Step 1

Introduction:

According to the question, X follows the log-normal distribution. P (X > 850) = 0.5; P (X > 100) = 0.9.

The natural logarithm (base e) of the log-normally distributed random variable with parameters µ and σ, has a normal distribution with mean µ and standard deviation σ. Let Y = ln (X).

Now, it is known that if loga(W) has a normal distribution, then logb(W) also has a normal distribution, as long as a, b > 0. However, the parameter values will change.

Step 2

Calculation:

Step 3

Calculation continued:...

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