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 )

{ 4 x 5 for x ≥ 1 0 otherwise

Solution

The given probability density function (PDF) is:

\begin{cases} \frac{4x}{5}, & x \geq 1 \\ 0, & \text{otherwise} \end{cases}$$ To calculate the variance and standard deviation, we follow these steps: ### 1. Normalize the PDF: We need to ensure that the total probability equals 1. Thus, we must check if the given PDF is normalized. $$\int_1^\infty f(x) \, dx = 1$$ Since $$f(x) = \frac{4x}{5}$$ for $$x \geq 1$$, calculate the integral: $$\int_1^\infty \frac{4x}{5} \, dx$$ If the integral is 1, the PDF is normalized. Otherwise, we'll need to adjust it. ### 2. Find the Expected Value (Mean) $$\mu = E[X]$$: $$E[X] = \int_1^\infty x f(x) \, dx$$ ### 3. Find the Expected Value of $$X^2$$: $$E[X^2] = \int_1^\infty x^2 f(x) \, dx$$ ### 4. Variance Formula: \[ \text{Variance} = \sigma^2 = E[X^2] - (E[X])^2 \] ### 5. Standard Deviation: $$\text{Standard Deviation} = \sigma = \sqrt{\text{Variance}}$$ Let's calculate these values step-by-step.from sympy import symbols, integrate, sqrt, oo # Define the variable and the PDF x = symbols('x') pdf = 4*x/5 # Calculate the normalization condition (integral of PDF from 1 to infinity) normalization_integral = integrate(pdf, (x, 1, oo)) # 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 normalization_integral_value = normalization_integral.evalf() mean_value = mean.evalf() mean_square_value = mean_square.evalf() variance_value = variance.evalf() std_deviation_value = std_deviation.evalf() normalization_integral_value, variance_value, std_deviation_valueIt appears the integral used for the normalization of the given PDF did not converge, leading to issues in calculating the variance and standard deviation. This suggests that the probability density function $$f(x) = \frac{4x}{5}$$ over the interval $$x \geq 1$$ is not properly normalized as it stands, resulting in the integral going to infinity. This could imply that either there is an issue with the function as specified or we may need to reconsider its bounds or definition. Would you like to clarify the problem, or should we try adjusting the function in some way?

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Math Problem Analysis

Mathematical Concepts

Probability Theory
Probability Density Function
Expected Value
Variance
Standard Deviation
Integral Calculus

Formulas

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

Theorems

-

Suitable Grade Level

Advanced Undergraduate

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