Given an exponential distribution with=10,what is the probability that the arrival time is a.less than X=0.2? b.greater than X=0.2? c.between X=0.2 and X=0.4? d.less than X=0.2 or greater than X=0.4? Round to four decimal places as needed.)
Question
Given an exponential distribution with=10,what is the probability that the arrival time is a.less than X=0.2? b.greater than X=0.2? c.between X=0.2 and X=0.4? d.less than X=0.2 or greater than X=0.4? Round to four decimal places as needed.)
Solution
The exponential distribution is defined as:
f(x|λ) = λ * exp(-λx) for x >= 0, 0 otherwise
Where λ is the rate parameter. The cumulative distribution function, which gives the probability that a random variable X following an exponential distribution is less than or equal to x, is given by:
F(x|λ) = 1 - exp(-λx)
Given λ = 10, we can calculate the probabilities as follows:
a. P(X < 0.2) = F(0.2|10) = 1 - exp(-10*0.2) = 1 - exp(-2) = 1 - 0.1353 = 0.8647
b. P(X > 0.2) = 1 - P(X < 0.2) = 1 - 0.8647 = 0.1353
c. P(0.2 < X < 0.4) = F(0.4|10) - F(0.2|10) = (1 - exp(-100.4)) - (1 - exp(-100.2)) = exp(-2) - exp(-4) = 0.1353 - 0.0183 = 0.1170
d. P(X < 0.2 or X > 0.4) = P(X < 0.2) + P(X > 0.4) = 0.8647 + (1 - F(0.4|10)) = 0.8647 + (1 - (1 - exp(-10*0.4))) = 0.8647 + exp(-4) = 0.8647 + 0.0183 = 0.8830
So, the answers are:
a. 0.8647 b. 0.1353 c. 0.1170 d. 0.8830
Please note that these are rounded to four decimal places.
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