Description
Exercise 1
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Generate 100 experiments of flipping 10 coins, each with 30% probability.
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What is the most common number? Why?
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Size= number of coin flips
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p= the probability of seeing one head in a coin flip
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Random variable denotes number of heads.
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~ 10, . 5
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Pr = 5 ?
Simulation:
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Repeat this experiment 100,000 times: “number of draws=100,000”
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flips <- rbinom(100000,10,.5)
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flips contains 100000 numbers, each between 0 and 10 (number of heads).
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mean(flips == 5), returns percentage of number “5” among 100000 numbers.
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dbinom(5,10,.5) returns probability of seeing 5 heads out of 10 tosses, for a fair coin using exact calculation.
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Note that if you re-run it, you will get the same result.
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As you can see, the result of exact calculation is 0.2460938 which is very close to the result of our simulation 0.24769
dbinom(k,10,.5) returns Pr = = ( )
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If you flip 10 coins each with a 30% probability of coming up heads, what is the probability exactly 2 of them are heads?
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Compare your simulation with the exact calculation.
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For exercise 2,
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Part a) use 10000 experiments and report the result.
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Part b) use 100000000 experiments and report the result.
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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.
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Simulation: run the experiment 100,000 times.
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flips <- rbinom (100000, 10, .5 )
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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.
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Simulation: run the experiment 100,000 times.
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flips <- rbinom (100000, 100, .2 )
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mean (flips): the average number of heads
Result of simulation is close to 20
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What is the expected value of a binomial distribution where 25 coins are flipped, each having a 30% chance of heads?
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Compare your simulation with the exact calculation.
If ~ 10, . 5 , what is the Var[ ]? using calculation Var =2.5.
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Simulation: run the experiment 100,000 times.
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X <- rbinom (100000, 10, .5 )
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var(X): the variance
Result of simulation is close to 2.5
If ~ 100, . 2 , what is the Var[ ]? using calculation Var = 16.
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Simulation: run the experiment 100,000 times.
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X <- rbinom (100000, 100, .2 )
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var(X): the variance
Result of simulation is close to 16
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What is the variance of a binomial distribution where 25 coins are flipped, each having a 30% chance of heads?
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Compare your simulation with the exact calculation.