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Problem 4 [11 points]. A random variable Y is said to have a log-normal distribution if X = ln(Y) has a normal distribution. The symbol ln(*) denotes the natural logarithm. The shape of the log-normal probability density function is similar to that of the gamma distribution, with a long tail to the right. The equation of the log-normal density function is given by: f(y) = { 1/(?y?2?) e^-((ln(y)-?)^2/(2?^2)), y > 0 0, elsewhere Because ln(y) is a monotonic function of y, P(Y ? y) = P(ln(Y) ? ln(y)) = P(X ? ln(y)) where X has a normal distribution with mean ? and variance ?^2. Thus, probabilities for random variables with a log-normal distribution can be found by transforming them into probabilities that can be computed using the ordinary normal distribution. If Y has a log-normal distribution with parameters ? = 4 and ?^2 = 1, a) (4 points) Apply the appropriate transformation tool to show that X = ln(Y) is indeed normally distributed. Identify the mean and variance for X. Hint: Refer to the method shown in class to derive the PDF of a random variable Y defined to be Y = X^2 in terms of the PDF of X for an idea to approach this.

          Problem 4 [11 points]. A random variable Y is said to have a log-normal distribution if X = ln(Y) has a normal distribution. The symbol ln(*) denotes the natural logarithm. The shape of the log-normal probability density function is similar to that of the gamma distribution, with a long tail to the right. The equation of the log-normal density function is given by:
f(y) = {
1/(?y?2?) e^-((ln(y)-?)^2/(2?^2)), y > 0
0, elsewhere
Because ln(y) is a monotonic function of y,
P(Y ? y) = P(ln(Y) ? ln(y)) = P(X ? ln(y))
where X has a normal distribution with mean ? and variance ?^2. Thus, probabilities for random variables with a log-normal distribution can be found by transforming them into probabilities that can be computed using the ordinary normal distribution. If Y has a log-normal distribution with parameters ? = 4 and ?^2 = 1,
a) (4 points) Apply the appropriate transformation tool to show that X = ln(Y) is indeed normally distributed. Identify the mean and variance for X. Hint: Refer to the method shown in class to derive the PDF of a random variable Y defined to be Y = X^2 in terms of the PDF of X for an idea to approach this.

Problem 4 [11 points]. A random variable Y is said to have a log-normal distribution if X = ln(Y) has a normal distribution. The symbol ln(*) denotes the natural logarithm. The shape of the log-normal probability density function is similar to that of the gamma distribution, with a long tail to the right. The equation of the log-normal density function is given by:
f(y) =
1/(?y?2?) e^-((ln(y)-?)^2/(2?^2)), y > 0
0, elsewhere
Because ln(y) is a monotonic function of y,
P(Y ? y) = P(ln(Y) ? ln(y)) = P(X ? ln(y))
where X has a normal distribution with mean ? and variance ?^2. Thus, probabilities for random variables with a log-normal distribution can be found by transforming them into probabilities that can be computed using the ordinary normal distribution. If Y has a log-normal distribution with parameters ? = 4 and ?^2 = 1,
a) (4 points) Apply the appropriate transformation tool to show that X = ln(Y) is indeed normally distributed. Identify the mean and variance for X. Hint: Refer to the method shown in class to derive the PDF of a random variable Y defined to be Y = X^2 in terms of the PDF of X for an idea to approach this.

Added by Travis K.

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