Introduction to Probability Theory

Computer Lab 4: The expectation of a random variable

In this lab, we will use large random samples as well as plots of probability functions and density functions to estimate the mean of common random variables.

1. The Bernoulli random variable
• Calculate the mean of a Bernoulli random variable with parameter p.
• Create a large random sample (use Calc -> Random Data -> Bernoulli...) of a Bernoulli random variable with p = 0.75 and find the sample mean (for instance with Stat -> Basic Statistics -> Display Descriptive Statistics... ) to check your result.
• Repeat the above for different values of p.
2. The binomial random variable

Consider a binomial random variable X with n = 5 and p = 0.75.
• Plot the probability function of this random variable
  • Enter the possible values of X in Column C1 and use Calc -> Probability distributions -> Binomial... to obtain the corresponding values of the probability function.
  • Plot the probability function of X.
  • Compute E(X) from the probability function of X.

• Estimate E(X) from a large random sample
  • Use Random Data -> Binomial... in the Calc menu to create a large random sample.
  • Use for instance Basic Statistics -> Display Descriptive Statistics... in the Stat menu to find the mean of your random sample.
  • Repeat the procedure with a larger random sample to make sure your estimate of E(X) is reasonable.

• Find an expression for E(X) in terms of n and p
  Repeat the above for different values of the parameters n and p of the random variable. Each time, record your estimate of E(X). Plot the estimate of E(X) as a function of the product n p. What do you observe?
3. The hypergeometric random variable
Recall that the random variable giving the number of "successes" when n drawings without replacement are made from a pool of N items, m of which are "successes" and N-m of which are "failures" is hypergeometric. Consider a hypergeometric random variable with N = 20, m = 15 and n = 6.
• Plot the probability function. of this random variable
  • Enter the possible values of X in one of the columns of the worksheet and use Calc -> Probability distributions -> Hypergeometric... to obtain the corresponding values of the probability function.
  • Plot the probability function of X.
  • Compute E(X) from the probability function. of X.

• Estimate E(X) from a large random sample
  • Use Random Data -> Hypergeometric... in the Calc menu to create a large random sample.
  • Use for instance Basic Statistics -> Display Descriptive Statistics... in the Stat menu to find the mean of your random sample.
  • Repeat the procedure with a larger random sample to make sure your estimate of E(X) is reasonable.

• Find an expression for E(X) in terms of m, n and N
  Repeat the above for different values of the parameters n and m of the random variable. Each time, record your estimate of E(X). Plot the estimate of E(X) as a function of the product n m. How are the results affected by changing the value of N ? Can you come up with an expression for E(X) in terms of m, n and N ?


The normal random variable
This continuous random variable is extremely important in probability theory because of the central limit theorem which, loosely speaking, states that many random phenomena empirically obey a normal distribution, at least approximately. A normal distribution is specified by two parameters, its mean m and its standard deviation s. The goal of this section is to explore how the density function changes when these two parameters are varied.

• Plot the density function of a normal random variable
  Use for instance m = 0 and s = 1. Make sure you generate enough data points: recall that this is a continuous random variable and that your graph of the density function should look continuous.

• Change the parameters and replot the density function
  • Change the mean but keep the standard deviation constant. What do you see?

    
    

  • Change the standard deviation while keeping the mean constant. What do you observe?