Problem 2. Suppose a particular kind of atom has a half-life of 1 year. This means that these atoms decay (transform) into another type of atom (or isotope). The lifetime of each single atom is assumed to have an exponential distribution with some rate > 0, which is determined by the half-life. The half-life is the time h by which the atom has a 1/2 probability of decaying. That is, if T is the survival time of an atom, P(T> h) = 1/2. Atoms in a large 'population' are assumed to decay independently. (1) Find by using that the half-life is 1 year. (2) Find the probability that an atom of this type survives at least 5 years. (3) Find the time at which the expected number of atoms is 10% of the original. Hint: use the indictor method with indicators of the form I,,t of the event that atom j has survived at least time t. (4) Find the probability that in fact none of the 1024 original atoms remains after the time calculated in (3).
Problem 2. Suppose a particular kind of atom has a half-life of 1 year. This means that these atoms decay (transform) into another type of atom (or isotope). The lifetime of each single atom is assumed to have an exponential distribution with some rate > 0, which is determined by the half-life. The half-life is the time h by which the atom has a 1/2 probability of decaying. That is, if T is the survival time of an atom, P(T> h) = 1/2. Atoms in a large 'population' are assumed to decay independently. (1) Find by using that the half-life is 1 year. (2) Find the probability that an atom of this type survives at least 5 years. (3) Find the time at which the expected number of atoms is 10% of the original. Hint: use the indictor method with indicators of the form I,,t of the event that atom j has survived at least time t. (4) Find the probability that in fact none of the 1024 original atoms remains after the time calculated in (3).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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Transcribed Image Text:Problem 2. Suppose a particular kind of atom has a half-life of 1 year. This means that these atoms
decay (transform) into another type of atom (or isotope). The lifetime of each single atom is assumed
to have an exponential distribution with some rate > 0, which is determined by the half-life. The
half-life is the time h by which the atom has a 1/2 probability of decaying. That is, if T is the survival
time of an atom, P(T> h) = 1/2. Atoms in a large 'population' are assumed to decay independently.
(1) Find by using that the half-life is 1 year.
(2) Find the probability that an atom of this type survives at least 5 years.
(3) Find the time at which the expected number of atoms is 10% of the original. Hint: use the
indictor method with indicators of the form I,,t of the event that atom j has survived at least
time t.
(4) Find the probability that in fact none of the 1024 original atoms remains after the time
calculated in (3).
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