Probability MomentsExp Exercises

Exercises

1. Find the number [math]z_{0}[/math] such that if [math]Z\sim N(0,1)[/math]

   1. [math]\Pr (Z\geq z_{0})=0.05[/math]

   2. [math]\Pr (Z\lt -z_{0})=0.025[/math]

   3. [math]\Pr (-z_{0}\lt Z\leq z_{0})=0.95[/math]

   and check your answers using EXCEL.

2. If [math]X\sim N(4,0.16)[/math] evaluate

   1. [math]\Pr (X\geq 4.2)[/math]

   2. [math]\Pr (3.9\lt X\leq 4.3)[/math]

   3. [math]\Pr \left( (X\leq 3.8)\cup (X\geq 4.2)\right) [/math]

   and check your answers using EXCEL. (Note for part (c), define the “events” [math]A=\left( X\leq 3.8\right) [/math] and [math]B=\left( X\geq 4.2\right) [/math] and calculate [math]\Pr \left( A\cup B\right)[/math].

3. 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].

4. 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].

5. 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]?