Final answer:
To determine the mean and variance of the lognormal random variable X, we use the formulas for mean and variance of a lognormal distribution. We can calculate P(X < 8) using the cumulative distribution function (CDF) of the lognormal distribution. The probability P(X < 0) from the lognormal distribution is zero, while the normal distribution can have non-zero probability for negative values.
Step-by-step explanation:
To determine the mean and variance of the lognormal random variable X, we can use the formulas for the mean and variance of a lognormal distribution. The mean of X is given by:
mean = e^(μ + σ^2/2)
where μ is the mean of the associated normal distribution and σ is the standard deviation of the associated normal distribution. In this case, the mean of the normal distribution is 1.5 and the standard deviation is 0.4. Therefore, the mean of X is:
mean = e^(1.5 + 0.4^2/2)
The variance of X is given by:
variance = (e^(σ^2) - 1) * e^(2μ + σ^2)
In this case, the variance of the normal distribution is 0.4^2 = 0.16. Therefore, the variance of X is:
variance = (e^(0.16) - 1) * e^(2 * 1.5 + 0.16)
P(X < 8) can be calculated using the cumulative distribution function (CDF) of the lognormal distribution. The CDF of the lognormal distribution is:
CDF(x) = P(X < x) = Φ((ln(x) - μ) / σ)
In this case, we can evaluate P(X < 8) using the given values of μ and σ.
To comment on the difference between the probability P(X < 0) calculated from the lognormal distribution and a normal distribution with the same mean and variance, we can compare the two probabilities. Since the lognormal distribution is only defined for positive values, the probability P(X < 0) is zero. On the other hand, the normal distribution can take negative values, so the probability P(X < 0) is not necessarily zero. Therefore, there is a difference between the two probabilities.