r - Area under probability density curve is not equal to 1 -

my model based on following growth function: x_t+1 = x_t + r * x_t * (1 – x_t/k) growth rate r defined each year random draw (mean=0.2, standard deviation=0.2). looking @ probability density function of stock x after 70 years 10000 runs. when calculate area under probability density function, equal 1 standard deviation of 0.1 not standard deviation of 0.2 or 0.3. created histogram area 100 values. shows high peak @ a=2 values area go 8. why not equal 1?

runs=10000  mean=0.2 sd=0.2  k=1 x_0=0.4 v=0.1 t=70  y=c() x=x_0   (j in 1:runs) {  rand=rnorm(t,mean,sd)     (i in 1:t) {       x=max(x+rand[i]*x*(x-v)*(1-x/k),0)       if(i==t)        y[j]=x     next}    x=x_0   next }    library(sfsmisc)    dens=density(y)    f=approxfun(dens$x, dens$y)    h=c()    i=seq(0.9, 1, length.out=100000)        (e in 1:length(i)) {          h[e]=f(i[e])       next}    options(max.print=1000000)     h[is.na(h)]=0    area=sum(abs(h[-1]+h[-length(h)])/2*diff(i)) 

thank you!

you numerical integration flawed.

first of all, plot of density:

almost of area underneath incredibly narrow, incredibly high spike. overwhelming majority of sample points outside of spike, hence don't job of estimating area of spike.

if evaluate print(dens) see this:

call:     density.default(x = y)  data: y (10000 obs.);   bandwidth 'bw' = 1.734e-06         x                y            min.   :0.9922   min.   :    0.0    1st qu.:0.9942   1st qu.:    0.0    median :0.9961   median :    0.0    mean   :0.9961   mean   :  439.1    3rd qu.:0.9981   3rd qu.:    0.0    max.   :1.0000   max.   :91131.2   

note in particular all of x values > 0.99, using numerical integration on interval [.9,1]. thus, of numerical integration outside of range of density. using (via approxfun) extrapolated values in range [0.9, 0.99], , extrapolating far outside (relatively speaking) range of data. 90% of numerical integration based on unreliable extrapolation of 2 almost-zero values (the first 2 y values).

you need find way rescale data. call density problematic. uses (as default) 512 points, if @ them, few dozen have y values greater 0. trying picture of distribution sampling tails rather values significant happening.