# Uniform distribution Part-II and Normal distribution Part-I

hello and welcome to todays lecture so we

would like to continue and discuss about two very important distributions to today but

before i start i would like to briefly review what was discussed in last class ok so in

last lecture we discussed about two distributions the poisson distribution ok so for the poisson

distribution ok probability of x equal to i is given by e to the power minus lambda

lambda to the power i by factorial i i is equal to zero one two and it goes up to infinity

ok so poisson distribution the poisson random variable is a discrete random variable ok

but it can go up to very high values ok and what we had ah demonstrated in last class

was for the poisson distribution your expectation and variance both return you a value of lambda

ok and so poisson distribution also one more thing ah so for large n and small p so by

small i mean lets say p is order point one so we can have the binomial random variable

is almost equivalent to the poisson random variable with parameter lambda is equal to

n times ok so other than the poisson distribution the

other distribution we discussed yesterday was the uniform uniform distribution ok and

for a uniform distribution ok in a range alpha to beta your so your probability is uniform

ok its a constant and this constant so this c is nothing but c is given by ok one by beta

minus alpha ok so probability of a less than x less than b is nothing but b minus a by

beta minus alpha ok so let us discuss some few very simple cases so imagine you have

a random variable ok x which is uniformly distributed over zero to ten so lets say you

want to calculate probability of two less than x less than five what is this probability

so this is nothing as you know from this example your b is equal to five a is equal to two

so this probability is nothing but five minus two and beta and alpha what is beta is ten

alpha is zero so ten minus zero is equal to three tenth ok

similarly probability of lets say one less than x less than four is nothing will return

your same value ok two less than x less than five and one less than x less than four both

will return you a value of three by ten what is probability of x greater than six x greater

than six is nothing but since you can go from six to ten so x greater than six would be

ten minus six by ten minus zero is equal to four by ten ok so you see how you can make

use of the uniform distribution to calculate each of the probabilities ok

let us take one more example ok so imagine at the bus stop ok your buses arrive every

fifty minutes so they arrive at seven a m seven fifteen a m and so on and so forth ok

so seven thirty a m and so on and so forth ok so the question is so if a person arrives

at a time which is uniformly distributed between ok between seven a m and seven thirty

a m what is the probability of waiting less than five minutes ok so let us read the problem

so the buses arrive at every fifteen minutes so seven seven fifteen seven thirty so on

and so forth ok so the person arrives at a time ok which is a random variable and which

is uniformly distributed between seven and seven thirty which means that person has equal

chance of arriving at seven seven one so on and ah seven one ah up to seven thirty ok

so what is the probability of waiting less than fifteen minutes ok less than five minutes

ok so since your buses arrive at seven ok seven

fifteen then seven thirty so if he has to wait less than five minutes ok he can arrive

anytime between ten to fifteen ok or seven ten to seven fifteen or anytime between seven

twenty five to seven thirty ok so what is my probability so this probability ok waiting

probability of waiting less than five minutes is probability of ten less than x less than

fifteen plus the probability of twenty five less than x less than thirty ok this is what

this is five divided by thirty why because you have the entire duration is thirty minutes

and this is also five divided by thirty so total is ten by thirty equal to one third

ok now we can also ask the question what is the

probability of waiting more than twelve minutes ok so the bus arrives at seven seven fifteen

seven thirty ok so if the person has to wait more than twelve minutes then he has to arrive

greater than seven to seven three right because if he arrives at four then he has to wait

only for eleven minutes ok since he has to wait at least ok more ok actually waiting

at least so this if it is more than twelve minutes then you have seven to seven two ok

or if ah also seven three also ok and similarly he can any anytime between fifteen right after

fifteen to eighteen ok c if he arrives just after fifteen then he would

have to wait for thirty minutes which is more than twelve minutes and latest he can arrive

at seven eighteen in that case you will have to wait for exactly twelve minutes ok so my

probability is nothing but probability of zero less than x less than three plus probability

of fifteen less than x less than eighteen equal to three by thirty plus three by thirty

equal to six by thirty nothing but one fifth ok so for a uniform distribution ok once again

i right this is one by beta minus alpha for alpha less than x less than beta and zero

otherwise ok you can calculate the expectation expectation of this variable is nothing but

x this will return you a value of ok ok

expectation would returning a value of beta minus alpha by two similarly i can calculate

expectation of x square ok so x cubed by three and x cubed is this so you will get a value

of sorry sorry i think i have made a mistake ok so let me redo this ok so expectation of

x ok is x square so one by beta minus alpha ok x square by two between alpha to beta equal

to beta square minus alpha square by two into beta minus alpha ok is equal to alpha plus

beta by two ok that makes sense on an average you will get a value which is average of alpha

and beta ok this also i will redo so expectation of x square is going to be one minus beta

minus alpha into x cube by three ok is beta cubed minus alpha cubed ok beta square plus

alpha beta plus alpha square by three into beta minus alpha so this would give you a

value of beta square plus alpha beta plus alpha square into by three ok

so your variance of x then becomes alpha square by three minus alpha plus beta by two whole

square ok you can simplify this and see what value you get ok so that brings our discussion

of uniform distribution to a close we will now discuss one of the most important distributions

which is the normal distribution ok ok so what is the normal distribution so normal

distribution is the mount shaped distributions that you see at all times ok this is your

x value this is your probability ok so for normal distribution ok your f of x is given

by ok ok so you see there are two parameters

here so your probability density function is given by f x and one by sigma root two

pi to the power minus x minus mu whole square by two sigma square ok so typically this peak

is roughly point four by sigma square ok this peak is roughly sorry point four by sigma

ok so we want to know what is this value of mu and sigma respectively ok

so for that as again we can use the moment generating function so for this is equal to

expectation of e to the power t x this will be one i can take out this is a constant ok

ok so so phi of t then becomes ok i can let us say i put this mu minus y minus mu by sigma

is equal to sorry so let me define y as x minus mu by sigma ok so phi of t becomes sigma

root two pi ok so if i put this then my d y is equal to d x by sigma ok so i can actually

convert it ok so i have minus two e to the power t x e to the power minus x minus mu

whole square two sigma square into d x right so in this would mean my d x is equal to sigma

into d y so limits wont change but if i put this values again so d x by sigma is equal

to d y so this sigma goes the limits remain unchanged power sigma y plus mu ok y square

by two into d y ok so i can take into t ok so i can take this

mu t term out ok because the integral is over y ok and i can write it as minus infinity

to infinity e to the power ok minus y square minus two sigma ok plus y by two into d y

right so we have minus y square two minus ok y square by two minus two sigma ok so this

you can so minus two sigma is what you get ok ok so this is plus ok so this i can write

as e to the power ok mu t by root two pi ok i can write minus infinity to infinity exponential

so within the exponential i can write it as y minus sigma t whole square by two plus sigma

square t square pi two ok into d y ok so i can take out this term again so i can

write it as e to the power mu t plus sigma square t square by two ok by ok root two pi

can be separate so this is exponential together into one by root two pi minus infinity to

infinity e to the power minus y minus sigma t whole square by two d y ok so this term

this term is nothing but the same distribution f of x with parameter of mu is equal to sigma

t and sigma is equal to one ok so this integral gives you nothing but one ok so i can then

write the entire phi of t is nothing but exponential mu t plus sigma square t square by two ok

so phi prime t then becomes simply mu plus sigma square t into exponential of the same

thing ok so phi prime zero is equal to expectation

of x which will be giving me nothing but mu if you put t equal to zero here this will

return you a value of zero this will return you a value zero so exponential of zero is

one this will return you a value of zero so only you get this mu so expectation of x is

equal to mu phi double prime of zero ok so first let us find out phi double prime of

t is equal to ok phi double prime of t is equal to sigma square exponential of mu t

plus sigma square t square by two so basically i have taken a derivative of this term keeping

this as constant and plus into mu plus sigma square t exponential of mu t plus sigma square

t square by two ok so this will return you a so phi double prime

of zero will give me a value of sigma square ok plus so t equal to zero you only get a

value of mu here also you get a value of mu and this whole thing is one ok so you get

a value of sigma square plus mu square so this is nothing but e of x square ok so variance

of x so this we have already opted is nothing but e of x square minus e x whole square is

equal to sigma square plus mu square ok sigma square plus mu square minus mu whole square

nothing but sigma square ok so you get variance of x is nothing but sigma ok

so for a normal distribution the mean is equal to mu and the variance is equal to sigma square

ok and what you can also observe that from this distribution ok from this distribution

ok since this is symmetric about the center ok so this is also the mode y because this

is the maximum occurring value and this is also the median ok because this is right because

this is the symmetric distribution and for every point to the left there is another point

to the right which is occurring at identical frequency so their average will always give

you this particular value ok which is nothing so this is nothing but mu ok so it means that

for a normal distribution so mean is equal to median is equal to mode

ok now for this if i know that f of x is defined as one by sigma root two pi e to the power

minus x minus mu whole square by two sigma square right so i can define this variable

z as x minus mu by sigma then f x ok so this f x i can then find out a cumulative distribution

function so this becomes f of z ok becomes ok one by root two pi e to the power minus

z square by two ok and the cumulative distribution function

ok of phi of x ok phi of x then becomes one by root two pi into minus infinity to infinity

e to the power minus y square by two d y ok minus infinity to x ok minus infinity less

than x less than infinity ok so this is your value so what is y phi of

x for this distribution ok for any value x phi of x is this area under the curve ok this

is your total probability of finding any x in any any value less than x ok so we can

write down for the normal distribution ok so for probability of x less than b is nothing

but probability of z less than b minus mu by sigma ok is equal to phi of b minus mu

by sigma ok so this is the value so these values these values of phi are all been computed

and you can find them from tables existing tables ok in any statistics book ok so i can

write down so if this is probability of x less than b ok as i drew earlier so for for

any b this is my probability of x less than b so i can find out for the z variable what

is the correspondence so this correspondence to b minus mu by sigma and accordingly i can

look this up value up ok i can write probability of a less than x less than b is nothing but

phi of b minus mu by sigma minus phi of a minus mu by sigma ok

so one more thing because of symmetry ok if this is your distribution ok so this is for

the entire x for phi actually you will have mean is always equal to zero right so for

phi mean is always equal to zero ok so you have minus infinity to infinity so for any

value x right so this area will also be equal to this area ok in other words so phi of minus

x so lets say this is value of x and this is minus x so this is your z axis right so

phi of minus x you know this area and this area are exactly small ok so i can write phi

of minus x is equal to one minus phi of x ok so this whole area is unity entire integration

will return you a value of unity and this area is nothing but whole area minus this

area and this come comes because of symmetry ok so accordingly by looking up these phi

values i can find out any particular value where x has a value less than equal to one

ok less than equal to infinity ok so let us just take one sample example ok

so if x is a normal random variable with mu is three and sigma square equal to sixteen

we want to calculate probability of x less than eleven so how do i calculate i just define

z is equal to x minus mu by sigma in this case it is x minus three by four ok so p of

x less than eleven is nothing but probability of z less than eleven minus three by four

is equal to phi of two ok so you can find out this is by looking up this value what

is the value of probability of x less than by looking up the table ok

let us take another example ok lets say you have gasoline used in compact cars generally

ok have mileage mileage of thirty five miles per gallon with standard deviation of four

point five miles per gallon so if a company wants to make a car that out performance ninety

five percent ok of cars in the market ok what must be the mileage ok so essentially we have

been asked to ask for what value of x naught ok for what value of x naught do we get probability

equal to point nine five ok so what do we do we define z naught as x naught minus thirty

five by four point five ok and find out ok so we want to find out for what value of z

naught by looking up the table what value of z naught do we get probability of z naught

is point ninety five ok once we know once we know that so for so for

example if you look up the normal distribution tables you will see that z naught is roughly

one point six four five so then you can use this equation to invert and find out what

is the value of x naught ok so that ah completes our discussion of normal distribution so today

we briefly discussed about some cases of uniform distribution and then we came and described

about normal distribution so for our normal distribution we derived that your expectation

is equal to mu and your variance is equal to sigma square ok so you can define us another

normal distribution ok where you define so you rescale the axis so you define z equal

to x minus mu by sigma then you will get a distribution which is centered at zero so

it will have mean of zero and standard deviation of ok of minus ok with that you can so you

can use then you can use this particular distributions to scale and because you have tables which

have these values you can transfer any translate any variable x into a standard normal variable

z ok and then look up the table to find out what exactly is the value ok with that

i thank you for your attention and we will meet again in next class