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2019-12-21 19:47:00
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$$ \frac{1}{n}\sum_{i=1}^{n}(a_i-\bar a)^2

\\

= \frac 1n \sum_{i=1}^{n}(a_i^2-2\bar a\times a_i+{\bar a}^2)

\\

= \frac 1n (\sum_{i=1}^nai^2 - 2\bar a\sum{i=1}^n a_i + n \bar a^2)

\\

= \frac 1n \sum_{i=1}^n a_i^2 - \bar a^2 $$

#include<bits/stdc++.h>
namespace ZDY{
    #pragma GCC optimize(3)
    #define il __inline__ __attribute__ ((always_inline))
    #define rg register
    #define ll long long
    #define ull unsigned long long
    #define db double
    #define sht short
    #define MB template <class T>il
    #define Fur(i,x,y) for(int i=x;i<=y;i++)
    #define Fdr(i,x,y) for(int i=x;i>=y;i--)
    #define fl(i,x) for(int i=head[x],to;to=e[i].to,i;i=e[i].nxt)
    #define clr(x,y) memset(x,y,sizeof(x))
    #define cpy(x,y) memcpy(x,y,sizeof(x))
    #define fin(s) freopen(s".in","r",stdin)
    #define fout(s) freopen(s".out","w",stdout)
    #define fcin ios::sync_with_stdio(false)
    #define l2(n) (int(log2(n)))
    #define inf 0x3f3f3f3f
    MB T ABS(T x){return x>0?x:-x;}
    MB T MAX(T x,T y){return x>y?x:y;}
    MB T MIN(T x,T y){return x<y?x:y;}
    MB T GCD(T x,T y){return y?GCD(y,x%y):x;}
    MB void SWAP(T &x,T &y){T t=x;x=y;y=t;}
}using namespace ZDY;using namespace std;
namespace IO{const char* ln="\n";const int str=1<<20;struct IN{char buf[str],*s,*t;bool _;IN():s(buf),t(buf),_(0){}il char gc(){return s==t&&((t=(s=buf)+fread(buf,1,str,stdin))==s)?EOF:(*s++);}IN&operator>>(char&ch){if(_)return *this;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF)_=1;else ch=c;return *this;}IN& operator>>(char* ch){if(_)return *this;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF)return _=1,*this;*ch=c;ch++;while((c=gc())!=EOF&&!isspace(c))*ch=c,ch++;if(c==EOF)_=1;return *this;}IN& operator>>(string& ch){if(_)return *this;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF)return _=1,*this;ch+=c;while((c=gc())!=EOF&&!isspace(c))ch+=c;if(c==EOF)_=1;return *this;}template<typename T>IN&operator>>(T&x){if(_)return *this;char c=gc();bool ff=0;while(c!=EOF&&(c<'0'||c>'9'))ff^=(c=='-'),c=gc();if(c==EOF){_=1;return *this;}x=0;while(c!=EOF&&'0'<=c&&c<='9')x=(x<<3)+(x<<1)+c-48,c=gc();if(c==EOF)_=1;if(ff)x=-x;return *this;}}in;struct OUT{char buf[str],*s,*t;OUT():s(buf),t(buf+str){}~OUT(){fwrite(buf,1,s-buf,stdout);}void pt(char c){(s==t)?(fwrite(s=buf,1,str,stdout),*s++=c):(*s++=c);}OUT&operator<<(const char*s){while(*s)pt(*s++);return *this;}OUT&operator<<(char*s){while(*s)pt(*s++);return *this;}OUT&operator<<(string s){for(int i=0;s[i];i++)pt(s[i]);return *this;}template<typename T>OUT&operator<<(T x){if(!x)return pt('0'),*this;if(x<0)pt('-'),x=-x;char a[30],t=0;while(x)a[t++]=x%10,x/=10;while(t--)pt(a[t]+'0');return *this;}}out;}using namespace IO;
#define N 100011
#define int ll
#define mod 1000000007
#define ls rt<<1
#define rs rt<<1|1
#define pu s[rt]=(s[ls]+s[rs])%mod,g[rt]=(g[ls]+g[rs])%mod
int n,q,s[N<<2],g[N<<2];
void build(int l,int r,int rt){
    if(l==r){
        in>>s[rt];g[rt]=(s[rt]*s[rt])%mod;
        return;
    }
    int m=(l+r)>>1;
    build(l,m,ls);build(m+1,r,rs);
    pu;
}
void upd(int x,int c,int l,int r,int rt){
    if(l==r){
        c%=mod;
        s[rt]=c;g[rt]=(c*c)%mod;
        return;
    }
    int m=(l+r)>>1;
    if(x<=m)upd(x,c,l,m,ls);
    else upd(x,c,m+1,r,rs);
    pu;
}
int rev(int x){
    int ans=1,p=mod-2;
    while(p){
        if(p&1)ans=(ans*x)%mod;
        x=(x*x)%mod;
        p>>=1;
    }
    return ans;
}
int ask(int L,int R,int v,int l=1,int r=n,int rt=1){
    if(L<=l&&r<=R)return (v?g[rt]:s[rt])%mod;
    int m=(l+r)>>1,ans=0;
    if(L<=m)ans=ask(L,R,v,l,m,ls);
    if(R>m)ans+=ask(L,R,v,m+1,r,rs);
    return ans%mod;
}
signed main(){
    in>>n>>q;
    build(1,n,1);
    int u,x,y;
    while(q--){
        in>>u>>x>>y;
        if(u==1)upd(x,y,1,n,1);
        else{
            int a=ask(x,y,0),A=ask(x,y,1),inv=rev(y-x+1);
            a=1ll*a*inv%mod;a=1ll*a*a%mod;
            out<<((1ll*A*inv%mod+mod)%mod-a+mod)%mod<<ln;
        }
    }
}
LG 5142 区间方差
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