$\"\"$

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(a)

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The exponential density function is $\"\"$ , where is $\"\"$ is some constant and $\"\"$ is in hours.

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Mean $\"\"$ .

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$\"\"$ .

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Thus, $\"\"$ .

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$\"\"$

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(i)

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Find the probability that a bulb fails within the first $\"\"$ , integrate $\"\"$ to $\"\"$ .

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$\"\"$

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Therefore, the probability that a bulb fails within the first $\"\"$ is $\"\"$ or $\"\"$ .

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$\"\"$

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(ii)

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Find the probability that a bulb burns for more than $\"\"$ .

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$\"\"$

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Therefore, the probability that a bulb burns for more than $\"\"$ is $\"\"$ or $\"\"$ .

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$\"\"$

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(b)

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Find the median lifetime of the lightbulbs.

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The median is at $\"\"$ , the probability from $\"\"$ to $\"\"$ (or $\"\"$ to $\"\"$ ) is $\"\"$ .

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Find the value of $\"\"$ by equating the integral to $\"\"$ .

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$\"\"$

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$\"\"$

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Take natural logarithm on both sides.

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$\"\"$

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Therefore, the median lifetime of the lightbulbs is $\"\"$ .

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$\"\"$

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(a)

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(i) $\"\"$ .

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(ii) $\"\"$ .

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(b)

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The median lifetime of the lightbulbs is $\"\"$ .