Probability RV Exercises

Probability Exercises

1. 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.)

   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.

2. 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];

   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];

   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].

   Check all your answers using EXCEL.

3. 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. 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].)

5. 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).