Let X ~ N(μ, σ²) and Y=eX(log-normal distribution). Find E[Y] and Var(Y).
Get the complete, step-by-step math solution for: "Let X ~ N(μ, σ²) and Y = e^X {(log-normal distribution)}. Find E[Y] and Var(Y).". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Define the probability density function of X
Since X follows a normal distribution with mean μ and variance σ2, its probability density function (PDF) is given by this formula. This PDF will be used to calculate the expected value of Y.
Step 2: Calculate E[Y] using the definition of expectation
The expected value of a function of a random variable, E[g(X)], is found by integrating g(x) multiplied by the PDF of X over its entire range. Here, g(X)=eX, so we substitute ex and the PDF of X into the integral.
Step 3: Simplify the exponent and complete the square
We combine the exponential terms and complete the square in the exponent. This involves algebraic manipulation to transform the expression into the form of a normal distribution's PDF. The integral of a PDF over its entire domain is 1, which simplifies the expression.
Step 4: Calculate E[Y]
After completing the square, the integral becomes the integral of a normal PDF with mean μ+σ2 and variance σ2. Since the integral of any PDF over its entire domain is 1, the expected value of Y simplifies to the exponential term outside the integral.
Step 5: Calculate E[Y²]
To find the variance, we first need to calculate E[Y2]. Since Y=eX, Y2=(eX)2=e2X. We can use the same method as for E[Y], but with 2X instead of X. This is equivalent to a normal distribution with mean 2μ and variance 4σ2.
Step 6: Apply the formula for E[e^(aX)]
Using the general result that if X∼N(μ,σ2), then E[eaX]=eaμ+2a2σ2, we substitute a=2. This gives us the value for E[Y2].
Step 7: Calculate Var(Y)
The variance of Y is defined as E[Y2]−(E[Y])2. We substitute the expressions we found for E[Y2] and E[Y] and simplify the result by factoring out common terms.