Probability MomentsExp Exercises - Revision history

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2014-09-05T15:22:20Z

‎Exercises

Rb at 15:17, 5 September 2014

2014-09-05T15:17:54Z

Rb at 13:33, 5 September 2014

2014-09-05T13:33:19Z

Rb at 12:06, 4 September 2014

2014-09-04T12:06:50Z

Rb: Created page with " = Exercises =

Find the number $z_{0}$ such that if $Z\sim N(0,1)$

$\Pr (Z\geq z_{0})=0.05$

2013-08-16T07:41:37Z

Created page with " = Exercises = <ol> <li>Find the number <math>z_{0}</math> such that if <math>Z\sim N(0,1)</math> <ol> <li><math>\Pr (Z\geq z_{0})=0.05</math></li> <li><m..."

New page

= Exercises =

<ol>
<li>Find the number <math>z_{0}</math> such that if <math>Z\sim N(0,1)</math>
<ol>
<li><math>\Pr (Z\geq z_{0})=0.05</math></li>
<li><math>\Pr (Z<-z_{0})=0.025</math></li>
<li><math>\Pr (-z_{0}<Z\leq z_{0})=0.95</math></li></ol>

and check your answers using EXCEL.</li>
<li>If <math>X\sim N(4,0.16)</math> evaluate
<ol>
<li><math>\Pr (X\geq 4.2)</math></li>
<li><math>\Pr (3.9<X\leq 4.3)</math></li>
<li><math>\Pr \left( (X\leq 3.8)\cup (X\geq 4.2)\right) </math></li></ol>

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>.</li>
<li>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>.</li>
<li>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>
<ol>
<li>Find the value of the constant, <math>k</math>, which ensures that this is a proper density function.</li>
<li>Evaluate <math>E[X]</math> and <math>var[X]</math>.</li></ol>
</li>
<li>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:
<ol>
<li>the simple average and variance of these <math>100</math> observations</li>
<li>the proportion of observations which are less than <math>1.</math></li></ol>

Now compare these with
<ol>
<li>the theoretical mean and variance of a <math>N\left( 2,1\right)</math> distribution</li>
<li>the probability that a random variable, with a <math>N\left( 2,1\right)</math> distribution, is less than <math>1</math>.</li></ol>

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>?</li></ol>

= Footnotes =

@@ Line 3: / Line 3: @@
 = 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 [ Q3].
+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].
 <ol>

@@ Line 1: / Line 1: @@
+Worked solutions to these questions can be found here: [http://youtu.be/UonenSstzlQ?hd=1 Q1], [http://youtu.be/olMokm6UFs0?hd=1 Q2] and [ Q3].
 <ol>
+<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>[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>.</p></li>
+<li><p><math>[L2]</math> 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, \\
 ,\quad \text{otherwise}.
@@ Line 14: / Line 13: @@
 \right.</math></p>
 <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><p>[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.</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>
 <ol>
@@ Line 28: / Line 27: @@
 <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> obervations, rather than just <math>100</math>?</p></li></ol>
+<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 =

@@ Line 7: / Line 7: @@
 <ol>
-<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>[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>.</p></li>
-<li><p>The continuous random variable <math>X</math> has probability density function given by</p>
+<li><p>[L2] 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, \\
 ,\quad \text{otherwise}.
@@ Line 17: / Line 17: @@
 <li><p>Evaluate <math>E[X]</math> and <math>var[X]</math>.</p></li></ol>
 </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>[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.</p>
 <p>Use EXCEL to calculate the following:</p>
 <ol>

@@ Line 6: / Line 6: @@
 <ol>
 <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>