Find the variance and standard deviation of the random variable X with the PDF given below. Enter the variance and then the standard deviation, separated by a comma and round to the nearest thousandth.f(x)={4x5forx≥10otherwise


To solve for the variance and standard deviation, let's break down the problem. The given probability density function (PDF) is:

\[
f(x) =
\begin{cases}
\frac{4x}{5}, & x \geq 1 \\
0, & \text{otherwise}
\end{cases}
\]

### Step 1: Normalize the PDF

The first step is to check if the PDF is normalized. We do this by ensuring that the total integral of the PDF over the range of \(x \geq 1\) equals 1:

\[
\int_1^\infty f(x) \, dx = 1
\]

### Step 2: Find the Mean (\( \mu = E[X] \))

The expected value (mean) of the random variable \(X\) is given by:

\[
E[X] = \int_1^\infty x f(x) \, dx
\]

### Step 3: Find \( E[X^2] \)

We also need to find \(E[X^2]\), which is calculated as:

\[
E[X^2] = \int_1^\infty x^2 f(x) \, dx
\]

### Step 4: Calculate the Variance and Standard Deviation

The variance is calculated as:

\[
\text{Variance} = \sigma^2 = E[X^2] - (E[X])^2
\]

Finally, the standard deviation is:

\[
\text{Standard Deviation} = \sigma = \sqrt{\text{Variance}}
\]

Now, let’s calculate these integrals and values.# Recalculating the integrals and handling the PDF correctly
from sympy import symbols, integrate, sqrt, oo

# Define the variable and the PDF
x = symbols('x')
pdf = 4*x/5

# Check the normalization of the PDF by integrating from 1 to infinity
normalization_integral = integrate(pdf, (x, 1, oo))

# If normalization is not 1, we can't proceed. But let's calculate
# Expected value (mean) E[X]
mean = integrate(x * pdf, (x, 1, oo))

# Expected value of X^2, E[X^2]
mean_square = integrate(x**2 * pdf, (x, 1, oo))

# Variance: E[X^2] - (E[X])^2
variance = mean_square - mean**2

# Standard deviation: sqrt(Variance)
std_deviation = sqrt(variance)

# Numerical values rounded to three decimal places
variance_value = variance.evalf()
std_deviation_value = std_deviation.evalf()

variance_value, std_deviation_valueIt seems there is a deeper issue with the given probability density function \( f(x) = \frac{4x}{5} \) for \( x \geq 1 \). The integrals are not converging, resulting in undefined values for the variance and standard deviation.

This suggests that the function as provided might not be a valid probability density function. For a proper PDF, the integral over the entire range must converge to 1, and the moments (such as \(E[X]\) and \(E[X^2]\)) should be finite. This PDF does not meet these criteria.

Could you confirm or adjust the PDF definition? If this is the correct function, we might need to reconsider how to proceed given the non-convergence.

Learn how to calculate the variance and standard deviation for a given probability density function (PDF). Use the provided PDF and step-by-step solution to find these statistical measures. Suitable for advanced undergraduate students studying probability theory.

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