Probability MomentsExp Exercises

Exercises

1. [L2] Suppose that [math]X[/math] is a Binomial random variable with parameters [math]n=3,[/math] [math]\pi =0.5,[/math] show by direct calculation that [math]E\left[ X\right] =1.5[/math] and [math]var \left[ X\right] =0.75[/math].

2. [L2] The continuous random variable [math]X[/math] has probability density function given by

   [math]f(x)=\left\{\begin{array}{c}0.1+kx,\quad 0\leq x\leq 5, \\ 0,\quad \text{otherwise}. \end{array} \right.[/math]

   1. Find the value of the constant, [math]k[/math], which ensures that this is a proper density function.

   2. Evaluate [math]E[X][/math] and [math]var[X][/math].

3. [L2] Use the random number generator in EXCEL to obtain [math]100[/math] observations from a [math]N\left( 2,1\right) [/math] distribution. When doing so, enter the last four digits from your registration number in the Random Seed field.

   Use EXCEL to calculate the following:

   1. the simple average and variance of these [math]100[/math] observations

   2. the proportion of observations which are less than [math]1.[/math]

   Now compare these with

   1. the theoretical mean and variance of a [math]N\left( 2,1\right)[/math] distribution

   2. the probability that a random variable, with a [math]N\left( 2,1\right)[/math] distribution, is less than [math]1[/math].

   What do you think would might happen to these comparisons if you were to generate [math]1000[/math] obervations, rather than just [math]100[/math]?