Solved-Binomial Distribution -Solution

Description

Exercise 1

• Generate 100 experiments of flipping 10 coins, each with 30% probability.

• What is the most common number? Why?

• Binomial Distribution has two parameters:

  • ~ ( , )

• • Size= number of coin flips

• • p= the probability of seeing one head in a coin flip

• • Random variable denotes number of heads.

• We flip a fair coins 10 times. What is the probability of seeing 5 heads?

• We flip a fair coins 10 times. What is the probability of seeing 5 heads?

Simulation:

• Repeat this experiment 100,000 times: “number of draws=100,000”

• flips <- rbinom(100000,10,.5)

• flips contains 100000 numbers, each between 0 and 10 (number of heads).

• mean(flips == 5), returns percentage of number “5” among 100000 numbers.

The result is 0.24769.

• dbinom(5,10,.5) returns probability of seeing 5 heads out of 10 tosses, for a fair coin using exact calculation.

• Note that if you re-run it, you will get the same result.

• As you can see, the result of exact calculation is 0.2460938 which is very close to the result of our simulation 0.24769

If ~ 10, . 5 , then

dbinom(k,10,.5) returns Pr = = ( )

Exercise 2

• If you flip 10 coins each with a 30% probability of coming up heads, what is the probability exactly 2 of them are heads?

• Compare your simulation with the exact calculation.

Exercise 3

• For exercise 2,

• Part a) use 10000 experiments and report the result.

• Part b) use 100000000 experiments and report the result.

• Compare the result of part a and part b, with the exact calculation. What is your conclusion?

If ~ 10, . 5 , what is the E[ ]? using calculation E = 5.

• Simulation: run the experiment 100,000 times.

• flips <- rbinom (100000, 10, .5 )

• mean (flips): the average number of heads

Result of simulation is close to 5

If ~ 100, . 2 , what is the E[ ]? using calculation E = 20.

• Simulation: run the experiment 100,000 times.

• flips <- rbinom (100000, 100, .2 )

• mean (flips): the average number of heads

Result of simulation is close to 20

Exercise 4

• What is the expected value of a binomial distribution where 25 coins are flipped, each having a 30% chance of heads?

• Compare your simulation with the exact calculation.

If ~ 10, . 5 , what is the Var[ ]? using calculation Var =2.5.

• Simulation: run the experiment 100,000 times.

• X <- rbinom (100000, 10, .5 )

• var(X): the variance

Result of simulation is close to 2.5

If ~ 100, . 2 , what is the Var[ ]? using calculation Var = 16.

• Simulation: run the experiment 100,000 times.

• X <- rbinom (100000, 100, .2 )

• var(X): the variance

Result of simulation is close to 16

Exercise 5

• What is the variance of a binomial distribution where 25 coins are flipped, each having a 30% chance of heads?

• Compare your simulation with the exact calculation.