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Problems

1. Suppose you have an unbiased coin, that is, it lands heads and tails with equal probability. You toss it 16 times and count the total number of heads you get. What's the standard deviation of the number of total number of heads. Now toss it 256 times. What's the standard deviation now?

2. Suppose you have the same setup as the last problem, with an unbiased coin. You give this coin to Bob so he can perform experiments on it. You want Bob to estimate the probability that a coin will land heads. Bob tosses it 16 times. He defines the probability of it landing heads to be $p =$ (total number of heads/total number of trials).

(a) On average, what result will Bob obtain?

(b) You want to estimate the standard deviation in Bob's result. Using the formula for the standard deviation of the binomial distribution, calculate the standard deviation of $p$ that Bob will ascertain.

(c) What happens to the standard deviation when Bob repeats the calculation 256 times instead?

• Contents
• Probability
  • Why Study Probability?
  • What is Probability?
    • Darwin awards
    • Not so fast, Charles
  • Calculation of Probability
    • Independence
    • Venn Diagrams
    • Bayes's Theorem
      • Cracking a lock
      • Back to Independence
    • Problems
    • Counting possibilities
      • Question
      • Question
    • Return to New York
      • Questions
    • Binomial Expansion
    • Binomial Distribution
      • Problems
    • Plotting the distro
    • Sharpness as N gets bigger
  • Continuous distributions
    • Problem
    • Infinitesimals in probability
    • Problem
  • Averaging
    • Gambling doesn't pay, sometimes
    • There are many ways to skin a guanabana
    • Averages of continuous variables
    • Properties of averages
    • Problems
    • Averages in involving multiple variables
      • Averages in involving multiple discrete variables
      • Averages in involving multiple continuous variables
      • Averages for independent variables
      • Problem
      • Average of a Binomial Distro
    • Variance: just how fat is your distribution?
      • *Variance of binomial distro
      • Problems
  • Gaussians are quite normal
    • Here there and everywhere
      • Problem
      • Was this a freak accident?
      • Variance of N variables
  • Random Walks
  • Correlations
    • Correlation vs. Causation
    • Linear Regression
  
  
• Statistics
  • Introduction
  • Simple Example
  • Unbiased estimates
  • Hypothesis testing
  • But what if I goofed?
  • Goofing up for fame and fortune
  • Forming a statistic
  • The shape of the distribution
    • Variance of mean differences
    • There's no such thing as a brontosaurus
  • How to do the t-test
  • Example
    • t-test data example
  • Can you beat Wall Street?
    • What are stocks?
    • Stock Trading: a market minute on NASDAQ
    • Bob's uncanny stock investment advice
      • Signal to noise
    • How random?
    • Deceptive Correlations
  • Stocks and Random Walks
    • Random Walks and Volatility
      • Volatility example
  • Stock beta calculation
  
  
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