Mean and Variance of Probability Density Function

Probability Density Function
The given probability density function is x π e 2 (x 4), which means that the probability of any value of x is directly proportional to the value of x multiplied by e to the power of negative 2 times the value of x 4.

Calculating Mean
To calculate the mean, we need to integrate the probability density function from negative infinity to infinity and multiply the result with x.

Mean (µ) = ∫-∞∞ x * (x π e 2 (x 4)) dx

= ∫0∞ x * (x π e 2 (x 4)) dx (since the function is even)

= ∫0∞ x^2 e^(-2x) dx

= 1/2 (1/2) = 1/4

Therefore, the mean of the probability density function is 1/4.

Calculating Variance
To calculate the variance, we need to integrate the squared difference of the probability density function and its mean from negative infinity to infinity.

Variance (σ^2) = ∫-∞∞ (x - µ)^2 * (x π e 2 (x 4)) dx

= ∫0∞ (x - 1/4)^2 * (x π e 2 (x 4)) dx

= ∫0∞ x^3 e^(-2x)/16 - x^2 e^(-2x)/2 + x e^(-2x)/4 - e^(-2x)/16 dx

= 3/32

Therefore, the variance of the probability density function is 3/32.

Conclusion
The mean of the probability density function is 1/4, and the variance is 3/32. The mean represents the expected value of the probability density function, whereas the variance represents the spread of the probability density function.