chance that File does handle resources, which I don't think it does, you'll have to convert the path to a URI first, for example using URI.create(). File(String) expects a path, not a URI.)

 Q:

distance from the origin in a simple random walk on $\mathbb Z^2$

let $S_{n}= \sum_{i=1}^{n}X_i$ be a simple random walk on $\mathbb{Z}$, with $S_0 = 0$.
$X_i = 1$ with probability $p$ and $X_i = -1$ with probability $1-p$.
It can be shown that 
$$\mathbb{P}(S_n=i\mid|S_n|=i, |S_{n-1}|=i_{n-1},..., |S_1|=i_1)= \frac{p^i}{p^i+(1-p)^i},$$
which means that having only $|S_n|=i$, one can calculate the probability of being in state $S_n=i$ (and therefore $S_n=-i$). From here, it is easy to show that $|S_n|$ is a markov chain.
I'm trying to show that the same is true for a (not necessarily symmetric) simple random walk  on $\mathbb{Z^2}$  starting from the origin. In other words, I want to show that distance from the origin at each step is a markov chain. 
The argument above does not seem to be extendable here, since there may be $k>1$ points with integer coordinates for a given distance from the origin and $k$ depends on the distance.

A:

This is not true. For instance, you have different options for the future whether you're at $(3,4)$ or $(5,0)$, and you can tell which one you're at by looking at the past, so the process is not memoryless.

 Q:

Lowercase all letters in a string except the first letter and capitalize first letter of string? -javascript

i.e If I have an input string:
input = 'hello World, whatS up?'
I want an output string:
desiredOutput = 'Hello World, whats up?'
If the first letter of any word in the string is already in Upper case, leave it as
chance that File does handle resources, which I don't think it do, you'll take to convert the path to a URI first, for exemplar using URI.create (). File(String) expects a way, not a URI .) 

  Q: 

 distance from the origin in a simple random walk on $ \mathbb Z^2 $ 

 permit $ S_{n}= \sum_{i=1}^{n}X_i$ be a simple random walk of life on $ \mathbb{Z}$, with $ S_0 = 0$. 
 $ X_i = 1 $ with probability $ p$ and $ X_i = -1 $ with probability $ 1 - p$. 
 It can be shown that 
 $ $ \mathbb{P}(S_n = i\mid|S_n|=i, |S_{n-1}|=i_{n-1 }, ..., |S_1|=i_1)= \frac{p^i}{p^i+(1 - p)^i},$$ 
 which means that take only $ |S_n|=i$, one can calculate the probability of being in state $ S_n = i$ (and therefore $ S_n=-i$). From here, it is easy to show that $ |S_n|$ is a markov range. 
 I'm trying to show that the same is dependable for a (not necessarily symmetric) elementary random walk   on $ \mathbb{Z^2}$   get down from the origin. In other words, I want to show that distance from the beginning at each step is a markov chain. 
 The argument above does not seem to be extendable here, since there may be $ k>1 $ points with integer coordinates for a given distance from the origin and $ k$ depends on the distance. 

 A: 

 This is not true. For instance, you have different options for the future whether you're at $ (3,4)$ or $ (5,0)$, and you can tell which one you're at by looking at the past, so the procedure is not memoryless. 

  Q: 

 Lowercase all letters in a string except the beginning letter and capitalize inaugural letter of string? -javascript 

 i.e If I have an input chain: 
 input =' hello World, whatS up?' 
 I want an output string: 
 desiredOutput =' Hello World, what s up?' 
 If the inaugural letter of any word in the string is already in Upper case, leave it as
chajce that File does handlt resources, which I don't think it does, you'll have to convert the peth ro a YRI first, for example jsing URI.breate(). Fioe(Stcing) expects a pefh, not a URI.)

 S:

fistcnre from the orinin in a sikple random wank ou $\mathbb Z^2$

let $S_{n}= \sum_{i=1}^{n}X_i$ be a simple random wwlk on $\mathbb{Z}$, witn $S_0 = 0$.
$S_p = 1$ with probability $p$ and $X_i = -1$ with pgobability $1-p$.
It cam be shown that 
$$\mathbb{P}(S_n=i\lid|S_j|=i, |S_{n-1}|=i_{n-1},..., |S_1|=i_1)= \frac{p^i}{p^l+(1-p)^i},$$
which meqns eyat having ovly $|S_n|=i$, ont ean calculaje the probability of being in sgate $X_n=i$ (and thgxwfogg $S_n=-i$). From hxre, it is easy to show thad $|S_n|$ is a markov chaik.
I'm tcyint to show that the sake is true for a (njt necessdrnly symmetric) simple eabdom falk  on $\natfbb{E^2}$  xtzrting frkm the orifin. In othee words, I want to snor that distancs from tre origin at each step is a markov chait. 
Tge argument above does bot seem to be extendwble here, since there may be $k>1$ points with integer coordindtes hof a glven eidtance from the origin and $k$ depends on the dyatsnbe.

A:

This is not trme. For instance, yoi jafg different opjions for fhe future whether you're wt $(3,4)$ oe $(5,0)$, and yot cam tell which one you're at bt looking at rhe past, so the prlcess is noc memotyless.

 Q:

Lowercase all letters in z string exfept the rkrst letter and zapptalhze firsu letter of string? -javascri't

i.e Nf I havd an input string:
inpkt = 'hcnlo World, whatS up?'
L wanj an ogtput strijg:
desiredOutput = 'Hello World, whevs up?'
If the forvt netter oy any eord in the sering is alreaby in Up'er care, leave in as
chance that File does handle resources, which think does, you'll to convert the for using URI.create(). File(String) a path, not URI.) Q: distance from the origin a simple random walk on $\mathbb Z^2$ let $S_{n}= \sum_{i=1}^{n}X_i$ be a simple walk on $\mathbb{Z}$, with $S_0 = 0$. $X_i = 1$ with probability $p$ $X_i -1$ probability It can be shown that $$\mathbb{P}(S_n=i\mid|S_n|=i, |S_{n-1}|=i_{n-1},..., |S_1|=i_1)= \frac{p^i}{p^i+(1-p)^i},$$ which means that having only $|S_n|=i$, one calculate the probability of being in state $S_n=i$ therefore $S_n=-i$). From here, is easy to show that is markov chain. trying show the same is for a (not necessarily symmetric) simple random walk on $\mathbb{Z^2}$ starting from the origin. In other words, want to distance from origin each is a markov argument above does not seem to since there may be $k>1$ points with integer for a distance from the origin and $k$ on the distance. A: This is not true. instance, you have different options for the future whether you're at $(3,4)$ or $(5,0)$, and tell which one you're by looking at past, the is memoryless. Q: all letters in a string except the first letter and capitalize letter of string? -javascript i.e If I have an input = World, whatS up?' want an output string: = World, whats up?' If letter any string already Upper case, leave it
chance that File does handle rEsources, whIch I dOn't ThiNk It doEs, yoU'll have to conveRT the Path to a URI first, for examPle usInG uRI.cREaTe(). FilE(String) EXpECTs a PaTh, Not A Uri.)

 Q:

DistaNce From the Origin in a sImpLe Random walk on $\MAtHbb Z^2$

let $S_{n}= \sUm_{i=1}^{N}X_i$ be a simple RanDom walK oN $\maTHbb{Z}$, wIth $s_0 = 0$.
$X_i = 1$ wiTh probABility $P$ and $X_i = -1$ witH pRObabilITy $1-p$.
It caN BE sHown That 
$$\mathbb{P}(S_n=i\mid|s_N|=i, |s_{N-1}|=i_{n-1},..., |S_1|=i_1)= \frac{p^i}{p^i+(1-p)^I},$$
which MeANs THAt hAviNg only $|S_n|=i$, oNe Can caLCulate tHE pROBAbiLIty of being in sTate $S_n=i$ (and tHEreFore $S_n=-I$). FRom HEre, it iS easy To SHow That $|S_n|$ is a maRkov Chain.
I'm trYing to SHow that THe same iS true fOr a (Not NeceSSaRiLy sYmMEtrIC) sImpLE raNdom walk  On $\MaThbb{Z^2}$  StarTING From The OrigIn. In oTher words, I wanT to Show THat DistaNce frOm thE oRigin At each Step iS a Markov chain. 
The aRgumEnt above dOes NoT seEm To be eXTendabLe hEre, Since thEre may bE $K>1$ poInTS WItH integer coordinateS fOR A gIven distAnce frOM tHe ORigin and $K$ dEpeNds oN THe disTancE.

a:

THis is not True. FoR InStAnce, you HaVe diffErEnt OptIons fOR the Future Whether yOu're aT $(3,4)$ Or $(5,0)$, and you can telL Which one you're AT bY LOoKIng aT thE past, so the pRoceSS is nOt meMOrYleSS.

 Q:

LowErcasE aLL lETters in a string excepT tHe firsT lettEr and capitaliZe first letTER Of string? -JavaSCrIPt

i.e If I have an iNput sTring:
input = 'HEllo WorlD, whats up?'
I want An output sTRIng:
desirEdOUtpUt = 'HEllO wOrLd, whats up?'
If thE FIrst LeTter of aNy wOrd in thE stRinG is AlrEaDy in Upper Case, leavE iT aS
chance that File does hand le resourc es, w hic h I  d on't  thi nk it does, yo u 'llhave to convert the pa th to  a URIf ir st, f or exam p le u sin gUR I.c re a te (). F ile (String ) expectsa p at h, not a URI . )
 Q:

dist anc e from the o rig in inasim p le ra ndo m wal k on $ \ mathbb  Z^2$

le t$ S_{n}= \sum_{i = 1 }^ {n}X _i$ be a simple r a nd o m walk on $\ma thbb{Z }$ ,  w i t h $ S_0  = 0$.
$X_ i= 1$w ith pro b ab i l i ty$ p$ and $X_i =  -1$ with p r oba bility  $ 1-p $ .
It c an be  s h own  that 
$$\m athb b{P}(S_n= i\mid| S _n|=i,| S_{n-1} |=i_{n -1} ,.. ., | S _1 |= i_1 )= \fr a c{ p^i } {p^ i+(1-p)^ i} ,$ $
whi ch m e a n s  tha t h avin g onl y $|S_n|=i$,one  can cal culat e the  pro ba bilit y of b eingin  state $S_n=i$(and  therefor e $ S_ n=- i$ ). Fr o m here , i t i s easyto show tha t$ | S _n |$ is a markov cha in . I' m trying  to sh o wth a t the sa me  is  tru e for a  (no t  n ecessari ly sym m et ri c) simp le  rando mwal k on $\ m athb b{Z^2} $  start ing f r om the origin. In other word s ,I wa n t to  sh ow that dis tanc e  fro m th e  o rig i n ateachst e pi s a markov chain. Th e argu mentabove does no t seem tob e extendab le h e re ,  since there m ay be  $k>1$ poi n ts withinteg er coord inates fo r a givendis tan cefro m th e origin and$ k $ de pe nds onthe  distan ce. 

A :
Thi sis not tr ue. Forin st an ce , y ou ha v e differ en t o pt ion s for the fu turewhet he ry ou' re at $ ( 3, 4 ) $ or  $ (5 ,0)$ , a nd  youcant ell  whichone you'r e a t  bylo ok ing atthe past, soth e processis  no t memo r y less.

Q:

Lowercase all lette r s in astr ing e xcep t the fir stletter  an d  capit alizefirst  l ett e r  of s t r in g?-j avascripti .eIf Iha ve a n input  string:
input = ' h ell o World, what S u p?'I wa nta no utp ut str i n g:
desiredOutpu t = 'Hello  W o rl d, whats u p ?'If  the fi rst let ter o f  any wo rd in the  string i salre a d y i n Upper ca se, leav e it as
chance that_File does_handle resources, which I_don't think_it_does, you'll_have_to convert the_path to a_URI first, for example_using URI.create(). File(String)_expects_a path, not a URI.)

 Q:

distance from the origin in a simple random walk_on_$\mathbb Z^2$

let_$S_{n}=_\sum_{i=1}^{n}X_i$_be a simple random walk_on $\mathbb{Z}$, with $S_0 =_0$.
$X_i =_1$ with probability $p$ and $X_i = -1$_with_probability $1-p$.
It can_be shown that 
$$\mathbb{P}(S_n=i\mid|S_n|=i, |S_{n-1}|=i_{n-1},..., |S_1|=i_1)= \frac{p^i}{p^i+(1-p)^i},$$
which means that_having only $|S_n|=i$, one can calculate_the probability of_being_in_state $S_n=i$ (and therefore_$S_n=-i$). From here, it is easy_to show that $|S_n|$ is a_markov chain.
I'm trying to show that the_same is true for a (not_necessarily symmetric) simple random walk_ on_$\mathbb{Z^2}$  starting from the_origin. In other_words, I_want to show_that distance from the origin at_each step is_a markov chain. 
The argument above_does_not seem to_be_extendable_here, since_there may be_$k>1$_points with_integer_coordinates for a given distance from_the_origin and $k$ depends on the distance.

A:

This_is not true. For_instance,_you have different options_for the future whether you're_at $(3,4)$ or $(5,0)$, and you_can tell_which one_you're at by looking at the past, so the process is_not memoryless.

 Q:

Lowercase all letters in_a string except the_first letter_and_capitalize first letter_of_string? -javascript

i.e_If I have an input string:
input =_'hello World,_whatS up?'
I want an output string:
desiredOutput_= 'Hello World, whats_up?'
If_the first letter of any word_in the string is already in_Upper case, leave it as