Continuous Random Variables: Practice 2 1.8 2008/06/13 14:19:55 GMT-5 2008/08/15 09:49:15.684 GMT-5 Barbara Illowsky illowskybarbara@deanza.edu Susan Dean deansusan@deanza.edu Connexions cnx@cnx.org continuous distribution elementary exercise exponential function graph homework practice probability random statistics variable In this module the student will explore the properties of data with an exponential distribution.

Student Learning Outcomes The student will explore the properties of data with a exponential distribution.

Given Carbon-14 is a radioactive element with a half-life of about 5730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121 . We start with 1 gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.

Describe the Data What is being measured here? Are the data discrete or continuous? Continuous In words, define the Random Variable X size 12{X} {} . X size 12{X} {} = Time (years) to decay carbon-14 What is the decay rate ( m size 12{m} {} )? m size 12{m} {} = 0.000121 The distribution for X size 12{X} {} is: X size 12{X} {} ~ Exp(0.000121)

Probability Find the amount (percent of 1 gram) of carbon-14 lasting less than 5730 years. This means, find P ( X < 5730 ) size 12{P \( X<"5730" \) } {} . aSketch the graph. Shade the area of interest.

bFind the probability. P ( X < 5730 ) size 12{P \( X<"5730" \) } {} = b P ( X < 5730 ) size 12{P \( X<"5730" \) } {} = 0.5001 Find the percentage of carbon-14 lasting longer than 10,000 years. aSketch the graph. Shade the area of interest.

bFind the probability. P ( X > 10000 ) size 12{P \( X<"5730" \) } {} = b P ( X > 10000 ) size 12{P \( X<"5730" \) } {} = 0.2982 Thirty percent (30%) of carbon-14 will decay within how many years? aSketch the graph. Shade the area of interest.

bFind the value k size 12{k} {} such that P ( X < k ) = 0 . 30 size 12{P \( X<k \) =0 "." "30"} {} . b k size 12{k} {} = 2947.73