Calls arrive at a call center at the rate of 42 per hour.What is the probability that the next call arrives in a.less than 2 minutes? b.more than 4 minutes? c.less than 1 minute? Round to four decimal places as needed.
Question
Calls arrive at a call center at the rate of 42 per hour.What is the probability that the next call arrives in a.less than 2 minutes? b.more than 4 minutes? c.less than 1 minute? 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 = 42 calls per hour, we need to convert this rate to calls per minute since the time intervals given in the question are in minutes. So, λ = 42/60 = 0.7 calls per minute.
We can calculate the probabilities as follows:
a. P(X < 2) = F(2|0.7) = 1 - exp(-0.7*2) = 1 - exp(-1.4) = 1 - 0.2466 = 0.7534
b. P(X > 4) = 1 - P(X < 4) = 1 - F(4|0.7) = 1 - (1 - exp(-0.7*4)) = 1 - (1 - exp(-2.8)) = 1 - (1 - 0.0608) = 0.0608
c. P(X < 1) = F(1|0.7) = 1 - exp(-0.7*1) = 1 - exp(-0.7) = 1 - 0.4966 = 0.5034
So, the answers are:
a. 0.7534 b. 0.0608 c. 0.5034
Please note that these are rounded to four decimal places.
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