Difference between revisions of "Probability MomentsExp Exercises"
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<li><p>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>.</p></li> | <li><p>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>.</p></li> | ||
<li><p>The continuous random variable <math>X</math> has probability density function given by</p> | <li><p>The continuous random variable <math>X</math> has probability density function given by</p> | ||
Revision as of 12:06, 4 September 2014
Exercises
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].
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]
Find the value of the constant, [math]k[/math], which ensures that this is a proper density function.
Evaluate [math]E[X][/math] and [math]var[X][/math].
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:
the simple average and variance of these [math]100[/math] observations
the proportion of observations which are less than [math]1.[/math]
Now compare these with
the theoretical mean and variance of a [math]N\left( 2,1\right)[/math] distribution
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]?
