Calculate Variance and Standard Deviation for Probability Density Function

Math Problem Statement

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

Solution

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

\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.

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Probability Density Function (PDF)
Expected Value
Variance
Standard Deviation

Formulas

Variance formula: Var(X) = E[X^2] - (E[X])^2
Standard deviation formula: Std(X) = sqrt(Var(X))

Theorems

Suitable Grade Level

Advanced Undergraduate

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