The problem is asking for the standard deviation of a random variable X, given its probability density function (pdf). The pdf is given as f(x) = 1/x for 1

Question

The problem is asking for the standard deviation of a random variable X, given its probability density function (pdf). The pdf is given as f(x) = 1/x for 1 < x < A, and 0 otherwise.

Here are the steps to solve this problem:

Step 1: Find the value of A
The total probability under a pdf must equal 1. Therefore, we need to find the value of A that makes the integral of f(x) from 1 to A equal to 1.

∫ from 1 to A of (1/x) dx = 1

This integral evaluates to ln(x) from 1 to A, which simplifies to ln(A) - ln(1). Since ln(1) = 0, we have ln(A) = 1. Solving for A gives A = e.

Step 2: Calculate the mean (μ)
The mean of a random variable X with pdf f(x) is given by the integral of x*f(x) over the range of X. In this case, the range is 1 to e.

μ = ∫ from 1 to e of (x * 1/x) dx = ∫ from 1 to e of dx = [x] from 1 to e = e - 1

Step 3: Calculate the second moment (E[X^2])
The second moment is given by the integral of x^2*f(x) over the range of X.

E[X^2] = ∫ from 1 to e of (x^2 * 1/x) dx = ∫ from 1 to e of x dx = 0.5*x^2 from 1 to e = 0.5*e^2 - 0.5

Step 4: Calculate the variance (σ^2)
The variance is given by the second moment minus the square of the mean.

σ^2 = E[X^2] - μ^2 = (0.5*e^2 - 0.5) - (e - 1)^2

Step 5: Calculate the standard deviation (σ)
The standard deviation is the square root of the variance.

σ = sqrt(σ^2)

Plug in the value you calculated for the variance and compute the square root to get the standard deviation.

Knowee AI · Accepted Answer