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Gaussians An important function in statistics is the
Chapter 7, Problem 75E(choose chapter or problem)
Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), \(f(x)=e^{-a x^{2}}\)
a. Graph the Gaussian for a = 0.5, 1, and 2.
b. Given that \(\int_{-\infty}^{\infty} e^{-a x^{2}} \ d x=\sqrt{\frac{\pi}{a}}\), compute the area under the curves in part (a).
c. Complete the square to evaluate \(\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} \ d x\) , where a > 0, b, and c are real numbers.
Questions & Answers
QUESTION:
Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), \(f(x)=e^{-a x^{2}}\)
a. Graph the Gaussian for a = 0.5, 1, and 2.
b. Given that \(\int_{-\infty}^{\infty} e^{-a x^{2}} \ d x=\sqrt{\frac{\pi}{a}}\), compute the area under the curves in part (a).
c. Complete the square to evaluate \(\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} \ d x\) , where a > 0, b, and c are real numbers.
ANSWER:Problem 75E
Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), .
a. Graph the Gaussian for a =0.5, 1, and 2.
b. Given that , compute the area under the curves in part (a).
c. Complete the square to evaluate , where a > 0, b, and c are real numbers.
Answer;
Step 1;
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Given gaussian function is; f(x) =
.
Now , we have to sketch the graph the Gaussian for a = 0.5, 1, and 2.
If a = 0.5 , then the graph of the Gaussian function y= is;
If a = 1 , then the graph of the Gaussian function y= is ;
If a = 2 , then the graph of the Gaussian function y= is ;
