Probability RV Exercises - Revision history

Rb: /* Probability Exercises */

2014-08-28T07:48:26Z

‎Probability Exercises

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2014-08-28T07:47:53Z

Rb at 22:34, 27 August 2014

2014-08-27T22:34:13Z

Rb: /* Probability Exercises */

2014-08-27T15:05:59Z

‎Probability Exercises

Rb at 15:04, 27 August 2014

2014-08-27T15:04:04Z

Rb: /* Probability Exercises */

2013-08-13T12:40:05Z

‎Probability Exercises

Rb: Created page with " = Probability Exercises =

In an experiment, if a mouse is administered dosage level $A$ of a certain (harmless) hormone then there is a $0.2$
2013-08-13T12:39:38Z

Created page with " = Probability Exercises = <ol> <li>In an experiment, if a mouse is administered dosage level <math>A</math> of a certain (harmless) hormone then there is a <math>0.2</ma..."

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= Probability Exercises =

<ol>
<li>In an experiment, if a mouse is administered dosage level <math>A</math> of a certain (harmless) hormone then there is a <math>0.2</math> probability that the mouse will show signs of aggression within one minute. For dosage levels <math>B</math> and <math>C</math>, the probabilities are <math>0.5</math> and <math>0.8</math>, respectively. Ten mice are given exactly the same dosage level of the hormone and, of these, exactly <math>6</math> shows signs of aggression within one minute of receiving the dose.
<ol>
<li>Calculate the probability of this happening for each of the three dosage levels, <math>A,B</math> and <math>C</math>. (This is essentially a Binomial random variable problem, so you can check your answers using EXCEL.)</li>
<li>Assuming that each of the three dosage levels was equally likely to have been administered in the first place (with a probability of <math>1/3</math>), use Bayes’ Theorem to evaluate the likelihood of each of the dosage levels ''given'' that <math>6</math> out of the <math>10</math> mice were observed to react in this way.</li></ol>
</li>
<li>Let <math>X</math> be the random variable indicating the number of incoming planes every <math>k</math> minutes at a large international airport, with probability mass function given by <math>p(x)=\Pr (X=x)=\frac{(0.9k)^{x}}{x!}\exp (-0.9k),\quad x=0,1,2,3,4,..</math>. Find the probabilities that there will be
<ol>
<li>exactly <math>9</math> incoming planes during a period of <math>5</math> minutes (i.e., find <math>\Pr (X=9)</math> when <math>k=5)</math>;</li>
<li>fewer than <math>5</math> incoming planes during a period of <math>4</math> minutes (i.e., find <math>\Pr (X<5)</math> when <math>k=4)</math>;</li>
<li>at least <math>4</math> incoming planes during an <math>2</math> minute period (i.e., find <math>\Pr (X\geq 4)</math> when <math>k=2)</math>.</li></ol>

Check all your answers using EXCEL.</li>
<li>The random variable <math>Y</math> is said to be ''Geometric'' if it has probability mass function given by <math>p(y)=\Pr (Y=y)=(1-\theta )\theta ^{y-1},\quad y=1,2,3,...;\quad 0<\theta
<1</math>; where <math>\theta </math> is an unknown ‘parameter’.
Show that the cumulative distribution function can be expressed as
<math>P(y)=\Pr (Y\leq y)=1-\theta ^{y},\quad y=1,2,3,...</math>
with <math>P(y)=0</math> for <math>y\leq 0</math> and <math>P(y)\rightarrow 1</math> as <math>y\rightarrow \infty. </math>
(Note that <math>P(y)=p(1)+p(2)+...+p(y)=\sum_{t=1}^{y}p(t)</math> can be written in longhand as
<math>P(y)=\left( 1-\theta \right) \left( 1+\theta +\theta ^{2}+\theta ^{3}+\ldots
+\theta ^{y-1}\right) .</math>
The term in the second bracket on the right-hand side is the sum of a ''Geometric Progression''.)</li>
<li>The weekly consumption of fuel for a certain machine is modelled by means of a continuous random variable, <math>X</math>, with probability density function
<math>g(x)=\left\{
\begin{array}{c}
3(1-x)^{2},\quad 0\leq x\leq 1, \\
0,\quad \text{otherwise}.\end{array}
\right.</math>
Consumption, <math>X</math>, is measured in hundreds of gallons per week.
<ol>
<li>Verify that <math>\int_{0}^{1}g(x)dx=1</math> and calculate <math>\Pr (X\leq 0.5)</math>.</li>
<li>How much fuel should be supplied each week if the machine is to run out fuel <math>10\%</math> of the time at most? (Note that if <math>s</math> denotes the supply of fuel, then the machine will run out if <math>X>s</math>.)</li></ol>
</li>
<li>The lifetime of a electrical component is measured in <math>100</math>s of hours by a random variable <math>T</math> having the following probability density function
<math>f(t)=\left\{
\begin{array}{c}
\exp (-t),\quad t>0, \\
0,\quad \text{otherwise}.\end{array}
\right.</math>
<ol>
<li>Show that the cumulative distribution function, <math>F(t)=\Pr (T\leq t)</math> is given by
<math>F(t)=\left\{
\begin{array}{ll}
1-\exp (-t), & t>0 \\
0 & t\leq 0.\end{array}
\right.</math></li>
<li>Show the probability that a component will operate for at least <math>200</math> hours without failure is <math>\Pr (T\geq 2)\cong 0.135</math>.?</li>
<li>Three of these electrical components operate independently of one another in a piece of equipment and the equipment fails if ANY ONE of the individual components fail. What is the probability that the equipment will operate for at least <math>200</math> hours without failure? (Use the result in (b) in a binomial context).</li></ol>
</li></ol>

= Footnotes =

@@ Line 3: / Line 3: @@
 = Probability Exercises =
-Solutions in {curly brackets}. Worked solution clips can be found here [http://youtu.be/GsHP3gMwqI8?hd=1 Q1], [http://youtu.be/4zSRlomN6q4?hd=1 Q2], [http://youtu.be/GnJ0_DEymxI?hd=1 Q3], [http://youtu.be/lFkoxN4QO0M?hd=1 Q4] and [http://youtu.be/4MwFJog0kSo?hd=1 Q5].
+Levels in [square brackets] and solutions in {curly brackets}. Worked solution clips can be found here [http://youtu.be/GsHP3gMwqI8?hd=1 Q1], [http://youtu.be/4zSRlomN6q4?hd=1 Q2], [http://youtu.be/GnJ0_DEymxI?hd=1 Q3], [http://youtu.be/lFkoxN4QO0M?hd=1 Q4] and [http://youtu.be/4MwFJog0kSo?hd=1 Q5].
 <ol>

@@ Line 3: / Line 3: @@
 = Probability Exercises =
-Solutions in {curly brackets}. Worked solution clips can be found here [http://youtu.be/GsHP3gMwqI8?hd=1 Q1], [ Q2], [http://youtu.be/GnJ0_DEymxI?hd=1 Q3], [http://youtu.be/lFkoxN4QO0M?hd=1 Q4] and [http://youtu.be/4MwFJog0kSo?hd=1 Q5].
+Solutions in {curly brackets}. Worked solution clips can be found here [http://youtu.be/GsHP3gMwqI8?hd=1 Q1], [http://youtu.be/4zSRlomN6q4?hd=1 Q2], [http://youtu.be/GnJ0_DEymxI?hd=1 Q3], [http://youtu.be/lFkoxN4QO0M?hd=1 Q4] and [http://youtu.be/4MwFJog0kSo?hd=1 Q5].
 <ol>

@@ Line 3: / Line 3: @@
 = Probability Exercises =
-Solutions in {curly brackets}. Worked solution clips can be found here [http://youtu.be/GsHP3gMwqI8?hd=1 Q1], [ Q2], [ Q3], and [ Q5].
+Solutions in {curly brackets}. Worked solution clips can be found here [http://youtu.be/GsHP3gMwqI8?hd=1 Q1], [ Q2], [http://youtu.be/GnJ0_DEymxI?hd=1 Q3], [http://youtu.be/lFkoxN4QO0M?hd=1 Q4] and [http://youtu.be/4MwFJog0kSo?hd=1 Q5].
 <ol>

@@ Line 52: / Line 52: @@
 \right.</math></p></li>
 <li><p>Show the probability that a component will operate for at least <math>200</math> hours without failure is <math>\Pr (T\geq 2)\cong 0.135</math>.?</p></li>
-<li><p>Three of these electrical components operate independently of one another in a piece of equipment and the equipment fails if ANY ONE of the individual components fail. What is the probability that the equipment will operate for at least <math>200</math> hours without failure? (Use the result in (5.2) in a binomial context).</p></li></ol>
+<li><p>Three of these electrical components operate independently of one another in a piece of equipment and the equipment fails if ANY ONE of the individual components fail. What is the probability that the equipment will operate for at least <math>200</math> hours without failure? (Use the result in (5.2) in a binomial context). {0.0025}</p></li></ol>
 </li></ol>
 = Footnotes =

@@ Line 2: / Line 2: @@
 = Probability Exercises =
 <ol>
-<li><p>In an experiment, if a mouse is administered dosage level <math>A</math> of a certain (harmless) hormone then there is a <math>0.2</math> probability that the mouse will show signs of aggression within one minute. For dosage levels <math>B</math> and <math>C</math>, the probabilities are <math>0.5</math> and <math>0.8</math>, respectively. Ten mice are given exactly the same dosage level of the hormone and, of these, exactly <math>6</math> shows signs of aggression within one minute of receiving the dose.</p>
+<li><p><math>[L1,L2]</math> In an experiment, if a mouse is administered dosage level <math>A</math> of a certain (harmless) hormone then there is a <math>0.2</math> probability that the mouse will show signs of aggression within one minute. For dosage levels <math>B</math> and <math>C</math>, the probabilities are <math>0.5</math> and <math>0.8</math>, respectively. Ten mice are given exactly the same dosage level of the hormone and, of these, exactly <math>6</math> shows signs of aggression within one minute of receiving the dose.</p>
 <ol>
-<li><p>Calculate the probability of this happening for each of the three dosage levels, <math>A,B</math> and <math>C</math>. (This is essentially a Binomial random variable problem, so you can check your answers using EXCEL.)</p></li>
+<li><p>Calculate the probability of this happening for each of the three dosage levels, <math>A,B</math> and <math>C</math>. (This is essentially a Binomial random variable problem, so you can check your answers using EXCEL.) {0.0055; 0.2051; 0.0881}</p></li>
-<li><p>Assuming that each of the three dosage levels was equally likely to have been administered in the first place (with a probability of <math>1/3</math>), use Bayes’ Theorem to evaluate the likelihood of each of the dosage levels ''given'' that <math>6</math> out of the <math>10</math> mice were observed to react in this way.</p></li></ol>
+<li><p>Assuming that each of the three dosage levels was equally likely to have been administered in the first place (with a probability of <math>1/3</math>), use Bayes’ Theorem to evaluate the likelihood of each of the dosage levels ''given'' that <math>6</math> out of the <math>10</math> mice were observed to react in this way. Most likely that dosis B was administered.</p></li></ol>
 </li>
-<li><p>Let <math>X</math> be the random variable indicating the number of incoming planes every <math>k</math> minutes at a large international airport, with probability mass function given by <math>p(x)=\Pr (X=x)=\frac{(0.9k)^{x}}{x!}\exp (-0.9k),\quad x=0,1,2,3,4,..</math>. Find the probabilities that there will be</p>
+<li><p><math>[L1,L2]</math> Let <math>X</math> be the random variable indicating the number of incoming planes every <math>k</math> minutes at a large international airport, with probability mass function given by <math>p(x)=\Pr (X=x)=\frac{(0.9k)^{x}}{x!}\exp (-0.9k),\quad x=0,1,2,3,4,..</math>. Find the probabilities that there will be</p>
 <ol>
-<li><p>exactly <math>9</math> incoming planes during a period of <math>5</math> minutes (i.e., find <math>\Pr (X=9)</math> when <math>k=5)</math>;</p></li>
+<li><p>exactly <math>9</math> incoming planes during a period of <math>5</math> minutes (i.e., find <math>\Pr (X=9)</math> when <math>k=5)</math>; { 0.02316}</p></li>
-<li><p>fewer than <math>5</math> incoming planes during a period of <math>4</math> minutes (i.e., find <math>\Pr (X<5)</math> when <math>k=4)</math>;</p></li>
+<li><p>fewer than <math>5</math> incoming planes during a period of <math>4</math> minutes (i.e., find <math>\Pr (X<5)</math> when <math>k=4)</math>; { 0.7065}</p></li>
-<li><p>at least <math>4</math> incoming planes during an <math>2</math> minute period (i.e., find <math>\Pr (X\geq 4)</math> when <math>k=2)</math>.</p></li></ol>
+<li><p>at least <math>4</math> incoming planes during an <math>2</math> minute period (i.e., find <math>\Pr (X\geq 4)</math> when <math>k=2)</math>. { 0.1087}</p></li></ol>
 <p>Check all your answers using EXCEL.</p></li>
-<li><p>The random variable <math>Y</math> is said to be ''Geometric'' if it has probability mass function given by <math>p(y)=\Pr (Y=y)=(1-\theta )\theta ^{y-1},\quad y=1,2,3,...;\quad 0<\theta
+<li><p><math>[L2]</math> The random variable <math>Y</math> is said to be ''Geometric'' if it has probability mass function given by <math>p(y)=\Pr (Y=y)=(1-\theta )\theta ^{y-1},\quad y=1,2,3,...;\quad 0<\theta
 <1</math>; where <math>\theta </math> is an unknown ‘parameter’.</p>
 <p>Show that the cumulative distribution function can be expressed as</p>
@@ Line 25: / Line 27: @@
 +\theta ^{y-1}\right) .</math></p>
 <p>The term in the second bracket on the right-hand side is the sum of a ''Geometric Progression''.)</p></li>
-<li><p>The weekly consumption of fuel for a certain machine is modelled by means of a continuous random variable, <math>X</math>, with probability density function</p>
+<li><p><math>[L2]</math> The weekly consumption of fuel for a certain machine is modelled by means of a continuous random variable, <math>X</math>, with probability density function</p>
 <p><math>g(x)=\left\{
 \begin{array}{c}
@@ Line 34: / Line 36: @@
 <ol>
 <li><p>Verify that <math>\int_{0}^{1}g(x)dx=1</math> and calculate <math>\Pr (X\leq 0.5)</math>.</p></li>
-<li><p>How much fuel should be supplied each week if the machine is to run out fuel <math>10\%</math> of the time at most? (Note that if <math>s</math> denotes the supply of fuel, then the machine will run out if <math>X>s</math>.)</p></li></ol>
+<li><p>How much fuel should be supplied each week if the machine is to run out fuel <math>10\%</math> of the time at most? (Note that if <math>s</math> denotes the supply of fuel, then the machine will run out if <math>X>s</math>.) Find the formula you need to solve and find an approximate solution. {approximately 0.54}</p></li></ol>
 </li>
-<li><p>The lifetime of a electrical component is measured in <math>100</math>s of hours by a random variable <math>T</math> having the following probability density function</p>
+<li><p><math>[L2]</math> The lifetime of a electrical component is measured in <math>100</math>s of hours by a random variable <math>T</math> having the following probability density function</p>
 <p><math>f(t)=\left\{
 \begin{array}{c}