Probability RV Exercises

Probability Exercises

Solutions in {curly brackets}. Worked solution clips can be found here Q1, [ Q2], [ Q3], and [ Q5].

1. [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.

   1. 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}

   2. 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.

2. [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

   1. 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}

   2. fewer than [math]5[/math] incoming planes during a period of [math]4[/math] minutes (i.e., find [math]\Pr (X\lt 5)[/math] when [math]k=4)[/math]; { 0.7065}

   3. 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}

   Check all your answers using EXCEL.

3. [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\lt \theta \lt 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.)

4. [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

   [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.

   1. Verify that [math]\int_{0}^{1}g(x)dx=1[/math] and calculate [math]\Pr (X\leq 0.5)[/math].

   2. 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\gt s[/math].) Find the formula you need to solve and find an approximate solution. {approximately 0.54}

5. [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

   [math]f(t)=\left\{ \begin{array}{c} \exp (-t),\quad t\gt 0, \\ 0,\quad \text{otherwise}.\end{array} \right.[/math]

   1. 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\gt 0 \\ 0 & t\leq 0.\end{array} \right.[/math]

   2. 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].?

   3. 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).