Solution Found!
If Y has a log-normal distribution with parameters ? and ?
Chapter 4, Problem 183SE(choose chapter or problem)
If Y has a log-normal distribution with parameters \(\mu\) and \(\sigma^{2}\), it can be shown that
\(E(Y)=e^{\left(\mu+\sigma^{2}\right) / 2} \quad \text { and } \quad V(Y)=e^{2 \mu+\sigma^{2}}\left(e^{\sigma^{2}}-1\right)\).
The grains composing polycrystalline metals tend to have weights that follow a log-normal
distribution. For a type of aluminum, gram weights have a log-normal distribution with \(\mu=3\) and \(\sigma=4\) (in units of \(10^{-2}\) g).
a Find the mean and variance of the grain weights.
b Find an interval in which at least 75% of the grain weights should lie. [Hint: Use Tchebysheff’s theorem.]
c Find the probability that a randomly chosen grain weighs less than the mean grain weight.
Equation Transcription:
Text Transcription:
mu and sigma^2
E(Y)=e(mu+sigma^2)/2 and V(Y)=e^2mu+sigma^2(e^sigma^2-1)
mu=3
sigma=4
10^-2
Questions & Answers
QUESTION:
If Y has a log-normal distribution with parameters \(\mu\) and \(\sigma^{2}\), it can be shown that
\(E(Y)=e^{\left(\mu+\sigma^{2}\right) / 2} \quad \text { and } \quad V(Y)=e^{2 \mu+\sigma^{2}}\left(e^{\sigma^{2}}-1\right)\).
The grains composing polycrystalline metals tend to have weights that follow a log-normal
distribution. For a type of aluminum, gram weights have a log-normal distribution with \(\mu=3\) and \(\sigma=4\) (in units of \(10^{-2}\) g).
a Find the mean and variance of the grain weights.
b Find an interval in which at least 75% of the grain weights should lie. [Hint: Use Tchebysheff’s theorem.]
c Find the probability that a randomly chosen grain weighs less than the mean grain weight.
Equation Transcription:
Text Transcription:
mu and sigma^2
E(Y)=e(mu+sigma^2)/2 and V(Y)=e^2mu+sigma^2(e^sigma^2-1)
mu=3
sigma=4
10^-2
ANSWER:
Solution:
Step 1 of 4:
It is given that Y denotes the grain weights and it has the Log-normal distribution with =3,
=4.
Also, it is given that E(Y)= and V(Y)=
.
Using this, we need to find the required values.