The random variable X has a log-normal distribution, when the mean of ln(X) = 5.45 and variance of ln(X) = 0.334,

what is the probability that X >139.76?

ln(139.76) = 4.94

So, just look up Z>(4.94-5.45)/0.334

To find the probability that X is greater than a given value, we need to use the properties of the log-normal distribution.

Step 1: Convert the mean and variance values to parameters of the underlying normal distribution.

The log-normal distribution is defined in terms of the parameters μ and σ^2, where μ is the mean and σ^2 is the variance of the natural logarithm of the random variable. To find these parameters, we use the following relationships:

μ = ln(mean of X)
σ^2 = ln(variance of X)

In this case, the mean of ln(X) is given as 5.45 and the variance of ln(X) is given as 0.334.

μ = ln(5.45) ≈ 1.691
σ^2 = ln(0.334) ≈ -1.097

Step 2: Convert the given value of X to its corresponding value in the log-normal distribution.

In this case, we want to find the probability that X is greater than 139.76. To convert this value to its corresponding value in the log-normal distribution, we use the following relationship:

X = e^(μ + σ^2/2)

X = e^(1.691 + (-1.097)/2)
X ≈ 41.17

Step 3: Calculate the probability that X is greater than the given value using the cumulative distribution function (CDF) of the log-normal distribution.

The CDF of the log-normal distribution gives us the cumulative probability up to a certain value. To find the probability that X is greater than 139.76, we subtract the CDF at 139.76 from 1.

P(X > 139.76) = 1 - P(X ≤ 139.76)
= 1 - CDF(139.76; μ, σ^2)

Step 4: Use a statistical software or a table to find the CDF of the log-normal distribution.

The calculation of the CDF of the log-normal distribution involves numerical techniques and is typically performed using statistical software or tables. You can use software such as R, Python, or Excel to calculate the probability.

Alternatively, you can also use online calculators or statistical software packages to calculate the probability directly.

Please note that the specific value of P(X > 139.76) will depend on the chosen software or table used for the calculation.

To find the probability that X > 139.76, we need to use the properties of the log-normal distribution. The log-normal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed.

To solve this problem, we will first need to calculate the parameters of the underlying normal distribution. The mean and variance of the logarithm of X are given as mean(ln(X)) = 5.45 and variance(ln(X)) = 0.334.

Step 1: Find the parameters of the underlying normal distribution
In a log-normal distribution, the mean and variance of the logarithm are related to the parameters of the underlying normal distribution as follows:

Mean of the logarithm, mean(ln(X)) = μ
Variance of the logarithm, variance(ln(X)) = σ^2

So, we have μ = 5.45 and σ^2 = 0.334.

Step 2: Calculate the parameters of the underlying normal distribution
From the mean and variance, we can determine the parameters of the underlying normal distribution as follows:

The mean of the underlying normal distribution is μ_n = ln((μ^2)/sqrt(μ^2 + σ^2))
The standard deviation of the underlying normal distribution is σ_n = sqrt(ln(1 + (σ^2)/(μ^2)))

Substituting the given values, we get:
μ_n = ln((5.45^2)/sqrt(5.45^2 + 0.334)) ≈ 1.118
σ_n = sqrt(ln(1 + (0.334)/(5.45^2))) ≈ 0.579

Step 3: Calculate the probability using the cumulative distribution function (CDF)
Once we have the parameters of the underlying normal distribution, we can calculate the probability using the cumulative distribution function (CDF) of the normal distribution.

The CDF gives us the probability that a random variable is less than or equal to a specific value. Since we want the probability that X is greater than 139.76, we subtract the CDF value from 1.

P(X > 139.76) = 1 - CDF(139.76)

Now, we need to calculate the CDF for the transformed normal distribution. We use the formula CDF(x) = Φ((ln(x) - μ_n) / σ_n), where Φ represents the cumulative distribution function of the standard normal distribution.

CDF(139.76) = Φ((ln(139.76) - 1.118) / 0.579)

Now, using statistical software, tables, or online calculators, we can find the corresponding value of Φ for the given argument and subtract it from 1 to find the probability.

Note: If you have access to a programming language like Python, you can use libraries like scipy.stats to calculate the probability directly.