%----------------------- lifetimecells.m ---------------------
close all force
clear all
disp('Lifetime of Cells')
% Cells in human body have a vast spread of life spans. One cell may last a day;
% another a lifetime. Red blood cells (RBCs) have a lifespan of several
% months and cannot replicate, which is the price RBCs pay for being
% specialized cells. The lifetime of a red blood cell can be modeled by
% an exponential distribution with density
% f(t) = 1/beta e^(-t/beta),
% where beta = 4 (in units of months). For simplicity, assume that
% when a particular cell dies it is instantly replaced by a newborn
% cell of the same type. For example, a replacement cell could be
% defined as any new cell born approximately at the time when the
% original cell died.
%
% (a) Find the expected lifetime of a single red blood cell.
% Find the probability that the cell's life exceeds 150
% days. Hint: Days have to be expressed in units of beta.
%------------------------sol (a)--------------------------
%Expected life: beta = 4 months
%150 days = 5 months
1-expcdf(5,4) %(Cells(a))
%ans = 0.2865
%%
% (b) A single cell and its replacements are monitored over the
% period of 1 year. How many deaths/replacements
% are observed on average?
% What is the probability that the number of deaths/replacements exceeds 5.
% Hint:
% Utilize a link between Exponential and Poisson distributions.
% In simple terms, if life-times are exponential with parameter
% beta, then the number of deaths/replacements in the time interval [0, t]
% is Poisson with parameter lambda= t/beta. Time units for
% t and beta have to be the same.
%----------------sol (b)-----------------
%1 year= 12 months; Poisson(12/4)
%Observed on average: expectation of Poisson(3) = 3.
%Or rationalize:
%12/average life time = 3, but this is informal
1-poisscdf(5, 3) %(Cells(b))
%ans = 0.0839
%%
% (c) Suppose that a single cell and two of its replacements are monitored.
% What is the distribution of their total lifetime? Find the probability
% that their total lifetime exceeds 1 year.
% Hint: Consult Gamma
% distribution. If n random variables are exponential with parameter
% beta then their sum is Gamma with
% parameters r = n and beta.
%--------------------- sol (c) ------------------------
1 - gamcdf(12, 3, 4) %(Cells(c))
%ans = 0.4232
%%
% (d) A particular cell is observed t = 2.2 months after its birth
% and is found to still be alive. What is the probability that the total
% lifetime of this cell will exceed 7.2 months?
% -------------------- sol (d) ------------------------
%(Cells(d)): By memoryless property
% P(X>=7.2|X>=2.2)=P(X>=7.2-2.2)=P(X>=5)
% which is equal to (Cells(a)), 0.2865
%