Question

Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²).
c. Complete the square to evaluate ∫ from -∞ to ∞ of e^(-(ax² + bx + c)) dx, where a > 0, b, and c are real numbers.

Accepted Answer

Start with the integral $$ \int_{-\infty}^{\infty} e^{-(ax^2 + bx + c)} \, dx $$, where $$a \u003e 0$$, and $$b, c$$ are real numbers. To simplify the exponent, complete the square for the quadratic expression $$ax^2 + bx + c. $$Factor out $$a$$ from the terms involving $$x$$: $$ax^2 + bx + c = a\left(x^2 + \frac{b}{a}x\right) + c. $$Next, complete the square inside the parentheses: $$x^2 + \frac{b}{a}x = \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2. $$Substitute this back to get $$ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c. $$Rewrite the integral using the completed square form: $$\int_{-\infty}^{\infty} e^{-\left[a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c\right]} \, dx = \int_{-\infty}^{\infty} e^{-a\left(x + \frac{b}{2a}\right)^2} e^{a\left(\frac{b}{2a}\right)^2 - c} \, dx. $$Since $$e^{a\left(\frac{b}{2a}\right)^2 - c} is a $$constant with respect to $$x$$, factor it out of the integral. Then, perform the substitution $$u = x + \frac{b}{2a}$$, which does not change the limits of integration because they are infinite. The integral reduces to $$e^{a\left(\frac{b}{2a}\right)^2 - c} \int_{-\infty}^{\infty} e^{-a u^2} \, du$$, which is a standard Gaussian integral.

Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²).
c. Complete the square to evaluate ∫ from -∞ to ∞ of e^(-(ax² + bx + c)) dx, where a > 0, b, and c are real numbers.

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Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²).c. Complete the square to evaluate ∫ from -∞ to ∞ of e^(-(ax² + bx + c)) dx, where a > 0, b, and c are real numbers.

Risposta video verificata per un problema simile:

Concetti chiave

Completing the Square

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Properties of Exponential Functions

Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²).
c. Complete the square to evaluate ∫ from -∞ to ∞ of e^(-(ax² + bx + c)) dx, where a > 0, b, and c are real numbers.