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?
• ~ 10, . 5
• Pr = 5 ?
• 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.
