Examples of Poisson Distribution, Business Mathematics and Statistics Video Lecture | Business Mathematics and Statistics - B Com

[Music]
hello friends today our topic is Poisson
distribution so let's see first the
formula formula is P of X equal to small
X that is equal to e raised to minus M
Ambridge to X upon X factorial now let's
see the meaning of the terms M m means
it can be a mean which is nothing but
average or the very important point M is
equal to n into P now n means the total
number whereas P means the probability
now is the third term that is X X means
the last line of the question because
that will be the condition like find
probability of something so it is the
last hand of the question
so our first example is a hospital
switchboard receives an average of four
emergency calls in ten minutes interval
what is probability that there are at
most two emergency calls so first we
have to find our M and our X we have to
find m and we have to find X here M
means either it can be average or it can
be n into P in the question they have
mentioned that average of four emergency
Falls
so M is 4 X is the last run of the
portion so find probability that there
are at most two emergency Falls
so X equal at most two means it cannot
be more than two it is less than or
equal to two so less than or equal to
two
now let's apply the Poisson formula P of
X equal to less than equal to 2 that is
equal to e raised to minus M M raised to
X upon X factorial so it is very simple
like it can be converted as P of X equal
to 0 plus P of X equal to 1 plus P of X
equal to 2 similarly you have to plug in
the values M remains same M is 4 then X
changes X will change like 0 1 & 2 then
you have to add all this and you get the
final answer so our second question our
second question is a transmission
channel has per digit error probability
P equal to 0.01 all right calculate
probability of more than one error in 10
receive digits so we have to solve this
now we are clear that here average is
not given so M is equal to NP why
because average is not even but n is
given all right n is 10 received
digits so n is 10 now probably it is
0.01 so first step is clear like finding
em now second step is to find what is
the question like we have to find X now
X is clearly mentioned more than one so
X is greater than one
okay now apply the formula now formula
is P of X is greater than 1 that is
equal to e raised to minus M NH 2 X upon
X factorial it can be written as 1 minus
P of X equal to 0 and plus X equal to 1
so just plug in the values like we know
for here X equal to 0 and M equal to
what will be this 10 into 1 by 100 that
is 0.1 so M is our point 1 all right so
it's very simple now like M is 0.1 X is
also here 1 and M is 0.1 just find the
values of this to add them and
subtracted 1 now the same sum can be
solved by binomial distribution also
why because n is given be also given and
q is nothing but 1 minus P so n is 10 P
is 0.01 then it will be 1 minus 0.01 so
now we have n PQ what is the formula of
binomial distribution n CX p h2 x q
raise to n minus x and our condition of
P of X is greater than 1 so similarly
here we have to convert this like P of X
equal to 0 plus P of X equal to 1 and
that should be subtracted by 1 alright
here we have n we have P we have Q and
here we have to solve differently like
here for X equal to 0 we have to solve
this and for X equal to 1 we have to
solve this so we can solve this by
binomial distribution as well now let's
talk about the property of Poisson
distribution the first property is
expected value that is represented as M
or it is also known as mean all right so
it is NP if n is not given then we have
to directly use M and that is average
second was variance variance is Sigma
square that is also M now this is unique
because in binomial distribution when we
talk about binomial mean or say expected
value G of X was NP but variance of X
was n PQ here you can see variance is M
so this is difference between both of
them all right then what is standard
deviation it is Sigma that is root of M
now suppose our question is if X follows
Poisson distribution such that P of X
equal 1 equal to 2 times P of X equal 2
fine mean and variance how we can find
this so first we know that Poisson
distribution formula Poisson
distribution formula is P of X equal
lines 30 that is e raised to minus M M
raised to X upon X factorial we can use
this use this and equate them so X is 1
so e raised to minus and M raised to 1
upon 1 factorial here X is 2 so 2 times
e raised to minus M M raised to 2 upon 2
factorial now we have equation between M
you can solve this and e raised to minus
M raised to minus M cancels 1 factorial
is 1 2 factorial is 2 so 2 2 cancels so
what is remaining M equal to L square
this indicates M square minus M equal to
0 this means M equal to 0 or 1 now M
cannot be equal to 0 therefore m equal
to 1 okay
now what is the question pine mean and
variance mean is equal to variance and
that is equal to M so our answer is one
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peace out