Question

Using R-studio

2. Consider an experiment where we flip a fair coin six times in a row, and i is the number of heads tossed:

            a.         Calculate the probability mass function for i = 0. . . 6 using the equation from Ross section 2.8 for Binomial Random Variables

            b.         Conduct a simulation of this experiment in R, with T trials of the experiment – pick several values of T from 10 to 10,000.

            c.         Create a plot of the theoretical result vs. your simulation at T = 100 and T = 10,000. Show that they converge as T increases.

Answer 1

Console Terminal x-sample (0:1,6,replace = TRUE) #will give 0 or 1 for 6 times where we use 1 as head and 0 as tails #so sumof this variable x gaives total number of heads > #So sum of this variable x gives total number of heads > sum(x) [1] 2 > X [1] 1 0 0 1 0 0 > #as we can see that in the above experiment we had 2 times head and 4 times tails which sum(x) also resonates > #a) Probability mass function for binomial >dbinom(0:6,6,prob = 0o. 5) #mass function using binomial [1] 0.015625 0.093750 0.234375 0. 3112 500 0.234375 0.093750 o.015625 > #So above is the probability mass function for i-0 ... 6

Console Terminal #b) function written below for I trials y={} > performexp-function (T) { for (i in 1:T) { x-sample (0:1,6,replace = TRUE) y[i]-sum(x) return(y) > performexp (10) [1] 3 5 4 4 454 3 1 2 table(performexp(10)) 1 2 3 4 5 1 5 1 2 1 > #i ad-2 above case we have performed experiment 10 times and found 1 times head -1 ,2 times head-5 , 3 times head-1,4ti me #same doing for random numbers from 10 to 10000 > table(performexp (5 55) ) 1 70 130 179 125 > table(perfor mexp (1000) ) 4 C 2 3 5 4 40 7 0 1 2 3 87 261 305 228 4 5 6 17 81 21 > table(perfor mexp ( 5000) ) 1 C 2 4 5 6 75 452 1171 1556 1170 484 92 > table(perfor mexp (9999) ) C 1 2 3 4 175 912 2363 3110 2327 944 168 > table(performexp (10000)) 1 948 2231 3188 2417 3 5 C 4 6 146 927 143 > ||

c)actuals plot for T=100

25 LO 1 2 3 5 6 0

theoretical plot for T=100

39 25 20 10 LO

actuals plot for T=10,000

6 5 4 3 2 1 0 LO 0000 0000 2500 009 000 009 C

Theoretical at T=10000 (If you cannot understand which plot it is then look at the Y axis and you will get it is for which T value)

0000 2500 0000 1500 000 009 0

R code for the above plots

#c) theoretical and actual compairision at T-100 > table(performexp (100) ) #actual result 1 2 4 5 6 14 22 28 26 7 barplot (table(performexp 100))) #actual plot barplot (dbinom (0:6,6,prob barplot (table (performexp 10000))) #actual plot @T-10000 barplot (dbinom (0:6,6,prob = 0.5) 10000) #Theoretical @T-10000 > 0.5) 100) >

As we can rightly observe that the plots start to converge as T increases from 100 to 100000 as the actuals vs theoretical difference starts becoming small in the later plots which is the tendency of Central Limit Theorem as well

Hope the above answer has helped you in understanding the problem. Please upvote the ans if it has really helped you. Good Luck!!