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The time (in minutes) for a lab assistant to prepare the equipment for a certain experiment is a random variable taking values between 25 and 35 minutes with p.d.f ,,otherwisef(x)={110,25≤x≤350 - Mathematics and Statistics

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Question

The time (in minutes) for a lab assistant to prepare the equipment for a certain experiment is a random variable taking values between 25 and 35 minutes with p.d.f 

`f(x) = {{:(1/10",", 25 ≤ x ≤ 35),(0",", "otherwise"):}`

What is the probability that preparation time exceeds 33 minutes? Also, find the c.d.f. of X.

Sum
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Solution

Required probability = P(X > 33)

= `int_33^∞ f(x) dx`

= `int_33^35 f(x) dx + int_35^∞ f(x) dx`

= `int_33^35 f(x) dx + 0`     ...[ f(x) = 0, when x > 35]

= `int_33^35 1/10 dx`

= `1/10 int_33^35 1dx`

= `1/10[x]_33^35`

= `1/10[35 - 33]`

= `2/10`

= `1/5`

Let F (x) be the c.d.f. of X

∴ F(x) = P[X ≤ x]

= `int_-∞^x f(x) dx`

= `int_-∞^25 f(x)dx + int_25^x f(x)dx`

= `0 + int_25^x f(x)dx`    ...[∵ f(x) = 0, when f(x) < 25]

= `int_25^x 1/10dx`

= `1/10 int_25^x 1dx`

= `1/10[x]_25^x`

= `1/10[x - 25]`

∴ F(x) = `(x - 25)/10`

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Probability Distribution of a Continuous Random Variable
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2012-2013 (March)

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