Lognormal distribution question

[TooTall65]

1. The lognormal distribution is identified as a model for concentration of a certain organic matter above a certain stretch of a highway. The parameters of this distribution are sigma = 1.9 and sigma = 0.9. a) Determine the expected value and the standard deviation of such distribution. b) The probability that the concentration of this compound will be at least 10.

2. I have absolutely no clue how to do this. My instructors lectures are next to useless and give pretty much no help.

3. See #2.

 

[andrewkirk]

If the concentration C has a lognormal distribution, that means that log C has a normal distribution. THe parameters $\mu$ and $\sigma$ are the mean and standard deviation of log C.

So the prob that C>10 is the prob that log C > log 10.

The wikipedia page on lognormal distributions is concise, correct and clear.

 

[TooTall65]

That does make sense, and thank you Andrew. I guess I'm confused when it asks for the 'expected value'. Expected value of what? I'm assuming it wants the mean, but it just says 'expected value and standard deviation'. It's confusing.

 

[Ray Vickson]

That does make sense, and thank you Andrew. I guess I'm confused when it asks for the 'expected value'. Expected value of what? I'm assuming it wants the mean, but it just says 'expected value and standard deviation'. It's confusing.

There should be no confusion---the terms 'expected value' and 'standard deviation (or variance)' apply to the random variable you are discussing, which is the lognormal random variable $X$ in this case. If you read carefully, you will see that the problem described the parameters as "sigma = 1.9 and sigma = 0.9" (but I guess that meant "mu = 1.9"). The problem did NOT use the terms mean (or expected value) and standard deviation to describe these parameters---although they are, in fact, the true mean and standard deviation of the related random variable $\ln X =Y \sim N(\mu, \sigma^2)$.

To summarize: you are given $\mu_Y = 1.9$ and $\sigma_Y = 0.9$, and from that you are asked to calculate $EX = \mu_X$ and $\sigma_X$.