Difference between revisions of "Probability MomentsExp Exercises"
(Created page with " = Exercises = <ol> <li><p>Find the number <math>z_{0}</math> such that if <math>Z\sim N(0,1)</math></p> <ol> <li><p><math>\Pr (Z\geq z_{0})=0.05</math></p></li> <li><p><m...") |
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= Exercises = | = Exercises = | ||
| − | + | Worked solutions to these questions can be found here: [http://youtu.be/UonenSstzlQ?hd=1 Q1], [http://youtu.be/olMokm6UFs0?hd=1 Q2] and [http://youtu.be/k9o8zKHl60Q?hd=1 Q3]. | |
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<ol> | <ol> | ||
| − | <li><p><math> | + | <li><p><math>[L2]</math> 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><math>[L2]</math> The continuous random variable <math>X</math> has probability density function given by</p> | |
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| − | <li><p>The continuous random variable <math>X</math> has probability density function given by</p> | ||
<p><math>f(x)=\left\{\begin{array}{c}0.1+kx,\quad 0\leq x\leq 5, \\ | <p><math>f(x)=\left\{\begin{array}{c}0.1+kx,\quad 0\leq x\leq 5, \\ | ||
0,\quad \text{otherwise}. | 0,\quad \text{otherwise}. | ||
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\right.</math></p> | \right.</math></p> | ||
<ol> | <ol> | ||
| − | <li><p>Find the value of the constant, <math>k</math>, which ensures that this is a proper density function.</p></li> | + | <li><p>Find the value of the constant, <math>k</math>, which ensures that this is a proper density function. Solution: 1/25</p></li> |
| − | <li><p>Evaluate <math>E[X]</math> and <math>var[X]</math>.</p></li></ol> | + | <li><p>Evaluate <math>E[X]</math> and <math>var[X]</math> 35/12 and 1.9097.</p></li></ol> |
</li> | </li> | ||
| − | <li><p>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.</p> | + | <li><p><math>[L2]</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.</p> |
<p>Use EXCEL to calculate the following:</p> | <p>Use EXCEL to calculate the following:</p> | ||
<ol> | <ol> | ||
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<li><p>the probability that a random variable, with a <math>N\left( 2,1\right)</math> distribution, is less than <math>1</math>.</p></li></ol> | <li><p>the probability that a random variable, with a <math>N\left( 2,1\right)</math> distribution, is less than <math>1</math>.</p></li></ol> | ||
| − | <p>What do you think would might happen to these comparisons if you were to generate <math>1000</math> | + | <p>What do you think would might happen to these comparisons if you were to generate <math>1000</math> observations, rather than just <math>100</math>?</p></li></ol> |
= Footnotes = | = Footnotes = | ||
Latest revision as of 15:22, 5 September 2014
Exercises
Worked solutions to these questions can be found here: Q1, Q2 and Q3.
[math][L2][/math] 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].
[math][L2][/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. Solution: 1/25
Evaluate [math]E[X][/math] and [math]var[X][/math] 35/12 and 1.9097.
[math][L2][/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] observations, rather than just [math]100[/math]?
