Question

In: Statistics and Probability

Using R-studio 2. Consider an experiment where we flip a fair coin six times in a...

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.

Solutions

Expert Solution

c)actuals plot for T=100

theoretical plot for T=100

actuals plot for T=10,000

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)

R code for the above plots

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!!


Related Solutions

We model the experiment where we toss a fair coin 4 times as follows. The sample...
We model the experiment where we toss a fair coin 4 times as follows. The sample space Ω is the space of (ordered) binary vectors of length 4. A 1 in the i-th position of such a binary vector indicates Heads at the i-th toss. The probability measure (P) on Ω is defined as P(A)=|A| / |Ω| for all A⊂Ω. Let Ai be the event that the ith toss is Heads, i = 1,...,4. (a) Give all outcomes in A3....
We flip a fair coin 20 times. Find the probability that we obtain between 8 and...
We flip a fair coin 20 times. Find the probability that we obtain between 8 and 17 heads, inclusively. Show work and please explain to someone that hardly understands statistics!
Assume we flip a fair coin 100 times. Use the normal approximation to the binomial distribution...
Assume we flip a fair coin 100 times. Use the normal approximation to the binomial distribution to approximate the probability of getting more than 60 heads. Answer: 0.0108 - need work
Suppose you have an experiment where you flip a coin three times. You then count the...
Suppose you have an experiment where you flip a coin three times. You then count the number of heads. State the random variable. Write the probability distribution for the number of heads. Draw a histogram for the number of heads. Find the mean number of heads. Find the variance for the number of heads. Find the standard deviation for the number of heads. Find the probability of having two or more number of heads. Is it unusual to flip two...
Flip a fair coin 4 times. Let ? and ? denote the number of heads and...
Flip a fair coin 4 times. Let ? and ? denote the number of heads and tails correspondingly. (a) What is the distribution of ?? What is the distribution of ? ? (b) Find the joint PMF. Are ? and ? independent? (c) Calculate ?(? ?) and ?(X≠?)(d) Calculate C??(?, ? ) and C???(?, ? )
1. Flip a fair coin ten times. Find the probability of at least seven heads. 2....
1. Flip a fair coin ten times. Find the probability of at least seven heads. 2. Draw five cards at once from a deck. Find the probability of getting two pairs. 3. Roll a die infinitely times. Find the probability that you see an even number before you see an one. 4. You and your friend take turns to draw from an urn containing one green marble and one hundred blue marbles, one at a time and you keep the...
USE R-studio TO WRITE THE CODES! # 2. More Coin Tosses Experiment: A coin toss has...
USE R-studio TO WRITE THE CODES! # 2. More Coin Tosses Experiment: A coin toss has outcomes {H, T}, with P(H) = .6. We do independent tosses of the coin until we get a head. Recall that we computed the sample space for this experiment in class, it has infinite number of outcomes. Define a random variable "tosses_till_heads" that counts the number of tosses until we get a heads. ```{r} ``` Use the replicate function, to run 100000 simulations of...
A fair coin is flipped six times. The outcomes of the coin flips form a palindrome...
A fair coin is flipped six times. The outcomes of the coin flips form a palindrome if the sequence of T’s and H’s reads the same forwards and backwards, e.g. THTTHT. Let A denote the event that the first, second and fourth flips are all ‘T’. Let Z denote the event that the six flips form a palindrome. (a) Is A independent of Z? (b) Is A independent of Z? (c) A fair coin flipped six times and a certain...
Flip a fair coin 5 times. What is the probability that at least one time the...
Flip a fair coin 5 times. What is the probability that at least one time the coin lands on heads?
Flip a fair coin 100 times. Let X equal the number of heads in the first...
Flip a fair coin 100 times. Let X equal the number of heads in the first 65 flips. Let Y equal the number of heads in the remaining 35 flips. (a) Find PX (x) and PY (y). (b) In a couple of sentences, explain whether X and Y are or are not independent? (c) Find PX,Y (x, y).
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT