PROBLEM

Generate 100 random numbers from Chi-square distribution with 12 degree freedom show  the histogram of this generated data and plot the corresponding density function on the same graph.

Theory-

The chi-squared distribution with df= n ≥ 0 degrees of freedom has density-

f_n(x) = 1 / (2^(n/2) Γ(n/2)) x^(n/2-1) e^(-x/2) ; x>0.

The mean and variance are n and 2n.

R Code-

#Command to remove previous objects
rm(list=ls())

#Generating 100 random numbers from Chi-Square Distribution
x=rchisq(100,12)

#Sorting the random numbers
x1=sort(x)

#Sorted random numbers are-
x1

#Histogram(prob. plot) of the generated random numbers
hist(x,prob=T,xlim=c(0,30),ylim=c(0,0.15),border="black",col="grey",breaks=15,
main = paste("Histogram and Density Plot of \nChi-Square Distribution"))

#Chi-Square Density Plot of Generated numbers
lines(x1,dchisq(x1,12),col="red",lwd=2)

#Drawing legend for the plot
legend("topright",legend=c(expression(bold("Histogram for 100 random no. from Chi-Square Dist.{df=12)")),
expression(bold("Density Plot for 100 ranom no. from Chi-Square Dist.{df=12)")))
,fill=c("grey","red"))

Output On R Console-

> #Command to remove previous objects
> rm(list=ls())
>
> #Generating 100 random numbers from Chi-Square Distribution
> x=rchisq(100,12)
>
> #Sorting the random numbers
> x1=sort(x)
>
> #Sorted random numbers are-
> x1
  [1]  3.563341  3.986165  4.790076  5.137254  5.475577  5.822501  5.866661
  [8]  6.010742  6.186136  6.306583  6.470758  6.596933  6.607481  6.624999
 [15]  6.670431  6.719123  6.721738  6.817978  7.048621  7.120953  7.472223
 [22]  7.784390  7.818839  7.862550  7.944896  7.945428  8.018721  8.222264
 [29]  8.348805  8.587792  8.602521  8.645715  8.725691  8.845125  8.909389
 [36]  9.099596  9.146373  9.192539  9.227033  9.369365  9.482021  9.649494
 [43]  9.696064  9.744548  9.765073  9.797461  9.874563  9.892526  9.946173
 [50] 10.005752 10.042422 10.051937 10.210783 10.293323 10.339786 10.350271
 [57] 10.414500 10.654812 11.020388 11.265752 11.704403 11.756806 11.887717
 [64] 12.306171 12.821964 13.033215 13.123094 13.264681 13.580782 13.595597
 [71] 13.662621 13.728855 13.838458 13.869891 13.975666 14.028286 14.064400
 [78] 14.616696 15.047347 15.185432 15.357018 15.740422 15.773002 15.871207
 [85] 16.021173 16.031287 16.034070 16.324973 17.933853 18.009785 18.030095
 [92] 18.591688 19.541009 19.547178 19.620247 21.427066 21.453698 21.716946
 [99] 21.722274 22.435368
>
> #Histogram(prob. plot) of the generated random numbers
> hist(x,prob=T,xlim=c(0,30),ylim=c(0,0.15),border="black",col="grey",breaks=15,
+ main = paste("Histogram and Density Plot of \nChi-Square Distribution"))
> 
>#Chi-Square Density Plot of Generated numbers
> lines(x1,dchisq(x1,12),col="red",lwd=2)
>
> #Drawing legend for the plot
> legend("topright",legend=c(expression(bold("Histogram for 100 random no. from Chi-Square Dist.{df=12)")),
+ expression(bold("Density Plot for 100 ranom no. from Chi-Square Dist.{df=12)")))
+ ,fill=c("grey","red"))

Output Graph-