$\"\"$

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The normal distribution is $\"\"$ .

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The probability function lies between two standard deviations $\"\"$ .

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Apply formula for the probability density function on $\"\"$ lies between $\"\"$ and $\"\"$ is $\"\"$ .

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The probability function lies between two standard deviations is $\"\"$ .

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$\"\"$

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Let $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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When $\"\"$ , then $\"\"$ .

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When $\"\"$ , then $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in integral.

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$\"\"$

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$\"\"$

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$\"\"$ .

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Using calculator $\"\"$ .

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$\"\"$

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$\"\"$ .

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The probability function lies between two standard deviations is $\"\"$ .

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$\"\"$

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The probability function lies between two standard deviations is $\"\"$ .