Question

The time required to repair a machine is exponentially distributed with parameter 0.5.

What is the probability that a repair time exceeds 2 hours? What is the conditional

probability that a repair time takes at least 10 hours given that its duration exceeds 9

hours?​

Answer 1

Answer:

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Answer 2

Answer:

Sol. Here parameter = k = 0.5

i. Probability that repair time exceeds 2 hours = P(X > 2)

= 1 − P(X < 2) =1-[ integral −∞ to 2( f(x)dx)]

= 1 − [∫−∞ to 0( f(x)dx)+ ∫0 to 2( f(x)dx)]

= 1 − [0 + ∫0 to 2( ke^−kxdx)]

= 1 − ∫0 to 2( 0.5e^−0.5xdx)

= 1 + [e^−0.5x]0 to 2

= 1 + e^−1 − e^0 = 1 + e−1 − 1 = 0.3678

ii. Conditional probability that a repair time takes at least 10 hours given that

its duration exceeds 9 hours

= P(X > 10/X > 9) =

P((X>10)∩(X>9))/P(X>9)

=P((X>10))/P(X>9)

=∫10 to ∞( f(x)dx)/ ∫9 to ∞ (f(x)dx )

=∫10 to ∞ (0.5e^−0.5xdx /∫9 to ∞ (0.5e^−0.5xdx )

=[e^−0.5x]10 to ∞ /[e^−0.5x]9 to ∞

=e−∞ − e−5/e−∞ − e−4.5

=0 − 0.006737 /0 − 0.011108 = 0.6065