use the package to verify your answer, using the commands var(X) and sd(X) and then by simply computing

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Exercise 2:

a) Sketch the probability mass function of X ∼ binom(size = 35, prob = 0.461).

b) Sketch the cumulative distribution function

c) Find the probability that X is equal to 21.

d) Find the probability that X is at most 18.

e) Find the probability that X is more than 30

f) Find the probability that X is between 12 and 24, inclusive. Solve the following using 3 methods as indicated and compare g) Find the mean of X: E(X), using the formula 𝜇𝜇 = ∑ 𝑥𝑥𝑥𝑥(𝑥𝑥) h) Install and use the package “distrEx” to verify your answer in question

g), with the command E(X) and then by simply computing n.p

i) Find the variance and standard deviation of X: 𝜎𝜎2 𝑎𝑎𝑎𝑎𝑎𝑎 𝜎𝜎, using the formula 𝜎𝜎2 = 𝐸𝐸(𝑋𝑋2) − �𝐸𝐸(𝑋𝑋)� 2 = 𝐸𝐸(𝑋𝑋2) − 𝜇𝜇2 (or equivalent)

j) Use the package to verify your answer, using the commands var(X) and sd(X) and then by simply computing npq =np(1-p).

Exercise 3: experiments with the binomial distribution Let X~binom(n = 100,000, p = 0.01), p = 0.01 probability of side effect resulting from a treatment and n = 100,000 treated individuals

a) Use your calculator to find 100000C110

b) Use R to find 100000C65

c) Use R to find 100000C125, then 100000C1100

d) Use dbinom to find f( 1100)

e) Use R to compute F(980)

f) Use the sum operator and dbinom to find P( 900 <= X<= 1200)

g) Compute F(1200)-F(900) using pbinom

. h) Use data.frame() to construct a table of x, F(x) for x = 980 to x = 1020

i) Use data.frame() to construct a table of x, f(x) for x = 980 to x = 1020

j) Graphing pmf f(x) Use the following code to graph the indicated binomial distribution for p= 0.1, 0.3, 0.5, 0.7 and 0.9. n <- 200 k 

Attachments:

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