Discrete probability distribution - Simulation

 

Simulate a random variable that counts the number of heads when four coins are tossed 64 times.
(a) Construct a frequency table
(b) Find the sample mean.
(c) Find the sample variance.

To generate the frequency table, use four random number generators to generate a random integer of 0 or 1 (0 for tails, 1 for heads). Then add the result. A possible frequency table is displayed below.

\[ \begin{align} x &| f_x \\ 0 &| 4 \\ 1 &| 16 \\ 2 &| 26 \\ 3 &| 12 \\ 4 &| 6 \\ \text{Total} &| 64 \end{align}\ \]

Use the definition to calculate the sample mean.

\[ \begin{align} \text{mean} &= \frac{\sum_{i=1}^n x_i \times f_i}{64} \\ &= \frac{0(4)+1(16)+2(26)+3(12)+4(6)}{64} \\ &= \frac{128}{64} \\ &= 2 \end{align}\ \]

Use the definition to find the sample variance

set mean = m, Then \[ \begin{align} s^2 &= \frac{\sum_{i=1}^n f_i \times (x_i - \text{m})^2 }{N-1} \quad \\ &= \frac{4(0 - 2)^2 + 16(1 - 2)^2 + 26(2 - 2)^2}{63} \quad \\ &+ \frac{12(3 - 2)^2 + 6(4 - 2)^2}{63} \quad \\ &= \frac{68}{63} \quad \\ &= 1.08 \end{align}\ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

 

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Now You Try:

Repeat the above using a random number generator from a spreadsheet or calculator.