The Student Room Group

Official TSR Mathematical Society

Scroll to see replies

A
Find limn2n1((1+1)1n(11)1n)\displaystyle\lim_{n\to \infty}\frac{2n}{\sqrt{-1}}\left((1+\sqrt{-1})^\frac{1}{n}-(1-\sqrt{-1})^\frac{1}{n} \right) .
Original post by bogstandardname

Spoiler




By eiπ=i2 \displaystyle e^{i\pi}=i^2, we obtain (1+i)1n(1i)1n=212n(eiπ4neiπ4n)(1+i)^{\frac{1}{n}}-(1-i)^{\frac{1}{n}}=2^{\frac{1}{2n}}(e^{\frac{i\pi}{4n}}-e^{-\frac{i\pi}{4n}}). Therefore, after some computations, we get
limn2n1((1+1)1n(11)1n)=limn212nsin(π4n)14n=π \displaystyle \lim_{n\to \infty}\frac{2n}{\sqrt{-1}}\left((1+\sqrt{-1})^\frac{1}{n}-(1-\sqrt{-1})^\frac{1}{n} \right)= \lim_{n\to \infty} 2^{\frac{1}{2n}}\frac{ \sin(\frac{ \pi}{4n})}{\frac{1}{4n}} = \pi.

n=145(1+tann)\displaystyle \prod_{n=1}^{45} \big(1+\tan n^\circ \big)
Original post by Lord of the Flies

n=145(1+tann)\displaystyle \prod_{n=1}^{45} \big(1+\tan n^\circ \big)


This is similar to a problem posted a while back. My solution was similar:
(1+tanx)(1+tan(45x))=2(1+tanx)(1+tan(45-x))=2
pairing up terms and multiplying by 1+tan45 yields 2^23
(edited 10 years ago)
Original post by Lord of the Flies

n=145(1+tann)\displaystyle \prod_{n=1}^{45} \big(1+\tan n^\circ \big)




abf(x)  dx=abf(a+bx)  dx\displaystyle \int_{a}^{b}f(x)\;{dx} = \int_{a}^{b}f(a+b-x)\;{dx}



akbf(k)=akbf(a+bk)\displaystyle \prod_{a \le k \le b}f(k) = \prod_{a \le k \le b}f(a+b-k)



akbf(k)=akbf(a+bk).\displaystyle \sum_{a \le k \le b}f(k) = \sum_{a \le k \le b}f(a+b-k).



Most people know the first one, but the discrete versions are just as handy. :]

(When applying it to this particular problem it's more convenient
to start the product from n=0n=0 -- that term being 11 anyway).
(edited 10 years ago)
Original post by ben-smith
This is similar to a problem posted a while back. My solution was similar: (1+tanx)(1+tan(45x))=2(1+\tan x)(1+\tan (45-x))=2

Pairing up terms and multiplying by 1+tan45 yields 2^23


Très bien!

Original post by L'art pour l'art
Most people know the first one, but the discrete versions are just as handy. :]

(When applying it to this particular problem it's more convenient to start the product from n=0n=0 -- that term being 11 anyway).


Indeed :biggrin:

I thought excluding the zero would make it a tad less obvious; but clearly the people on TSR are too good. :rolleyes:
Given that limxf(x)xγ=l0\displaystyle \lim_{x\to \infty} \frac{f(x)}{x^{\gamma}}=l \neq 0, a>0\displaystyle a > 0, 0<α<β\displaystyle 0 < \alpha < \beta, γ>0\displaystyle \gamma > 0, find limnv=1nf(vαanβ)\displaystyle \lim_{n\to \infty} \sum_{v=1}^{n} f(\frac{v^{\alpha}a}{n^{\beta}}).
please go on my new FM thread, i really need serious advice on this

dydx=sinx(cosx+y)cosxy\displaystyle \dfrac{dy}{dx}=\dfrac{sinx(cosx+y)}{cosx-y}
Hi. Can someone explain the Bionormial equation to me . i know that Binormial means the probability of success over failure from what i have read and that the probability of success in each trial is the same threfore- we only use inormal is we want to fidn the probability of success of failure. in order to find the bionormal distribution we have to know the
n- total population
r- number of choices chosen
p- success
q-failure

and the equation is (this is what i type into the calculator) nCrqn-rpr (currently, i am just memorising this quation meaning that i dont know what is the purpose of C (combination) being in the bionormial question .

so far, i am doing alright by asnwering a few binormial question until i came across this one:


2. A particularly long traffic light on your morning commute is green 20% of the time that you approach it. Assume that each mornign represents an independent trial.

b) Over the next 20 mornings, what is the probability that the light is green on exactly four days?


my solution:
20C4x(o.8)^16x(0.2)^4 + 20C3x(o.8)^17x(0.2)^3 + 20C2x(o.8)^18x(0.2)^2 + 20C1x(o.8)^19x(0.2)^1

now this is wrong. the correct answer is : 20C4x(o.8)^16x(0.2)^4 = 0.218.

can someone please explain? i need to learn my statistics asap as my exam is by the end of the month
Original post by lingwin1993
Hi. Can someone explain the Bionormial equation to me . i know that Binormial means the probability of success over failure from what i have read and that the probability of success in each trial is the same threfore- we only use inormal is we want to fidn the probability of success of failure. in order to find the bionormal distribution we have to know the
n- total population
r- number of choices chosen
p- success
q-failure

and the equation is (this is what i type into the calculator) nCrqn-rpr (currently, i am just memorising this quation meaning that i dont know what is the purpose of C (combination) being in the bionormial question .

so far, i am doing alright by asnwering a few binormial question until i came across this one:


2. A particularly long traffic light on your morning commute is green 20% of the time that you approach it. Assume that each mornign represents an independent trial.

b) Over the next 20 mornings, what is the probability that the light is green on exactly four days?


my solution:
20C4x(o.8)^16x(0.2)^4 + 20C3x(o.8)^17x(0.2)^3 + 20C2x(o.8)^18x(0.2)^2 + 20C1x(o.8)^19x(0.2)^1

now this is wrong. the correct answer is : 20C4x(o.8)^16x(0.2)^4 = 0.218.

can someone please explain? i need to learn my statistics asap as my exam is by the end of the month


Welcome to TSR :smile:
You want to find the probability that the variable occurs exactly 4 times - not the probability of it happening less than or equal to 4 times, which is what you found.
The better way to do this if you needed to would be to use the binomial tables found in the formula book.
Original post by joostan
Welcome to TSR :smile:
You want to find the probability that the variable occurs exactly 4 times - not the probability of it happening less than or equal to 4 times, which is what you found.
The better way to do this if you needed to would be to use the binomial tables found in the formula book.


and why is there a C (combination formula) in the binormial equation ?
Original post by lingwin1993
and why is there a C (combination formula) in the binormial equation ?


I'd be careful using terms like binomial equation, unless you mean this: http://planetmath.org/binomialequation :tongue:
The variable is distributed binomially
The C as you've correctly pointed out, is the combinations formula.
You are interested in combinations - i.e the exact number of values. The P or permutations includes all the combinations, but also expresses the different ways in which you can arrange the values given too.
Remember:
nCr=n!r!(nr)!^n\mathrm{C}_r = \frac{n!}{r!(n-r)!} and this is what you want to find.
Original post by joostan
I'd be careful using terms like binomial equation, unless you mean this: http://planetmath.org/binomialequation :tongue:
The variable is distributed binomially
The C as you've correctly pointed out, is the combinations formula.
You are interested in combinations - i.e the exact number of values. The P or permutations includes all the combinations, but also expresses the different ways in which you can arrange the values given too.
Remember:
nCr=n!r!(nr)!^n\mathrm{C}_r = \frac{n!}{r!(n-r)!} and this is what you want to find.


Are we allowed to ask for explaination on a mathemathical term here in this forum? Because i have tried reading up the defition and the function of "the poisson distribution" and i have to say it is not easy to grasp.

Mind helping? how does the Poitsson distribution worK? and what does it mean? so far i know that it is similiar to Binormial and that it has this "lumda" which this Lumda refer to 2 prenominals?
Original post by lingwin1993
Are we allowed to ask for explaination on a mathemathical term here in this forum? Because i have tried reading up the defition and the function of "the poisson distribution" and i have to say it is not easy to grasp.

Mind helping? how does the Poitsson distribution worK? and what does it mean? so far i know that it is similiar to Binormial and that it has this "lumda" which this Lumda refer to 2 prenominals?


I'm puzzled by what you are saying. . .
The poisson distribution has probability mass function defined as:

 Pr(X=k)=λkeλk!\ Pr(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}

Where λ=Var(X)=E(X) \lambda = Var(X)=E(X) and X is a stochastic variable
Btw its pronounced lambda :smile:
(edited 10 years ago)
Original post by joostan
I'm puzzled by what you are saying. . .
The poisson distribution has probability mass function defined as:

 Pr(X=k)=λkeλk!\ Pr(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}

Where λ=Var(X)=E(X) \lambda = Var(X)=E(X) and X is a stochastic variable
Btw its pronounced lambda :smile:


Could you please give me clues for this question? i dont get this question so well and i assume this question is about Poisson distribution

4) the number of surface flaws in plastic panels used in the interior of automobiles has a poisson distriution with a mean of 0.05 flaws per square foot of plastic panel. Assume an automobile interior contains 10 square feet of plastic panel.

a) what is the probability that there are no surface flaws in an auto's interior?


hm.. after i read it twice i thought i need to ask for help
hello Euler.
Original post by lingwin1993
Could you please give me clues for this question? i dont get this question so well and i assume this question is about Poisson distribution

4) the number of surface flaws in plastic panels used in the interior of automobiles has a poisson distriution with a mean of 0.05 flaws per square foot of plastic panel. Assume an automobile interior contains 10 square feet of plastic panel.

a) what is the probability that there are no surface flaws in an auto's interior?


hm.. after i read it twice i thought i need to ask for help


So: λ=0.05\lambda = 0.05
Sub this in, as well as a value of 0 for k, to the equation I have already given you.
Does Variance and Lambda have the same meaning? I am confused
Original post by lingwin1993
Does Variance and Lambda have the same meaning? I am confused


If you quote me I'm more likely to spot it :smile:
λ=Var(X)\lambda = Var(X) Where X is the variable. λ\lambda is just a Greek Letter :smile:

Quick Reply

Latest