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प्रश्न
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.
उत्तर
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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