Question

We want to explore the effect of parameters on a binomial distribution (this is a bit similar to last week's assignment). Start up R/RStudio

a) First we'll do our coin example from lecture. Let's calculate those probabilities and plot a graph so we know what our “distribution” looks like: In the script window or on the command line do the following: y <- c(0:10) we're giving y the numbers 0 through 10 y if you want to see “y”, you can do this step (not necessary) pr <- dbinom(y,10,.5) this gives us all the probabilities for the binomial. “.5” is the probability of success, “10” is the number of trials, and y is the number we're interested in (in this case, all the values from 0 to 10). Make sure you put in the numbers in the right order. pr this should give you all the probabilities. barplot(pr,names.arg = y) this generates a plot of “pr” (our probabilities) and tells it to use “y” to label our x axis. Now answer the question at the end of this section.

Now answer the question below: (a) What does the graph look like? (It should be identical to the one some of you may have seen in lecture). Make sure you present your graph as well. Instructions for cutting and pasting graphs were given in the previous homework assignment.

b) Let's try some different “parameters”. Let's change .5 to .1 and see what happens (kind of like having a coin that comes up heads 90% of the time): Repeat the above, but put use .1 for “probability of success” for both the graph and actual probabilities (for both the command line and R-commander, you essentially do the same thing you did, but change .5 to .

1). Now answer the question:

(b) What does the graph look like now? What changed? Why? Which probabilities are now higher? Why? Again, make sure you present your graph.

Answer 1

Here we see and infer from the graph that it is perfectly symmetrical in shape. If a normal plot is drawn by joining the points of the tips of the bars of the bargraph we may get a bell-shaped structure in the graph

However when we change the value of the parameter of the binomial distribution from 0.5 to 0.1 we see a a positively skewed nature graph which is presented to us. Here,success refers to the no. of tails coming up in the 10 trials that occurs. Now since the probability of a tail coming up is 10% that is, 10% of the times tails come up and 90% of the times , heads, it is understandable that the probability of lesser number of tails in 10 trials will be much higher in nature(since the coin is biased in nature) and hence we see and understand from the graph that the probability of only 1 tail coming up from 10 tossings of a coin is a much more probable event than the event of 4 tails or 5 tails coming up in 10 tossings. Also in comparison with the previous graph, we can see that when the coin is unbiased, that is, it is equally likely for both heads and tails to come up in 10 tossings the most probable event or the event having the most probability was of 5 tails from 10 tossings, whereas in this graph, the probability of the event of 5 tails coming up from 10 tossings is least likely to occur