Deck 6: Continuous Random Variables

Question 1

Let X be a random variable with uniform density function on . Find the density function of Z = X² .

Answer 1

Z = X² has density $f ( z ) = \frac { 1 } { 2 \sqrt { z } }$ on (1,9) and f(z)=0 otherwise. For c,d∈(1,3), $$P \left( c \leq X ^ { 2 } \leq d \right) = P ( \sqrt { C } \leq X \leq \sqrt { d } ) = \frac { \sqrt { d } - \sqrt { c } } { b - a } = \int _ { c } ^ { d } f ( z ) d z .$$

Question 2

Suppose X is a non-negative random variable with probability density function $f ( x ) = x e ^ { - x }$ Find P(2≤X)..

Answer 2

$P ( 2 \leq X ) = \int _ { 2 } ^ { \infty } x e ^ { - x } d x = \frac { 1 } { e ^ { 2 } }$

Question 3

Find the constant so that $$f ( x ) = \left\{ \begin{array} { l l } 0 & x < - 1 \\\frac { x ^ { 2 } } { 3 } & - 1 \leq x \leq A \\0 & x > A\end{array} \right.$$ is a probability density function.

Answer 3

A=2.

Question 4

Let X be a continuous random variable with probability density function $$f ( x ) = \left\{ \begin{array} { l l } \frac { x ^ { 3 } } { 4 } & 0 \leq x \leq 2 \\0 & \text { else }\end{array} \right.$$ Find the probability density function of f(x)=2eˣ.

Question 5

A lightbulb company knows that its lightbulbs have a lifespan (in thousands of hours) modelled by a random variable with the probability density function $\left\{ \begin{array} { l l } \frac { A } { ( 2 - x ) ^ { 4 } } & 3 < x < 4 \\0 & \text { else. }\end{array} . \right.$ .
(a) Find
(b) If a certain company installs 3 of their lightbulbs at the same time, what is the probability that all three need to be replaced within 3500 hours?

Question 6

Let X be a random variable with probability density $f ( x ) = \left( 2 e ^ { 2 } \right) e ^ { - 2 x }$ for x>1. Find Var(X).

Question 7

Let Z be a random point in the interal (0,π/2). . Find the probability density function of Y=sin(Z).

Question 8

A dentist has found that a certain procedure has a cost (in thousands of dollars) with probability density function: $$f ( x ) = \left\{ \begin{array} { l l } 2 e ^ { - 2 t } & x \geq 0 \\0 & \text { else }\end{array} \right.$$ . The dentist also found that if she charges the patients 10% of the cost up front, then k, the amount of money the insurance company pays, only in 12% of the time exceeds the remaining cost. Find k.

Question 9

Let Z be a continuous random variable taking values in $( - \pi / 2 , \pi / 2 )$ with distribution function and probability density function f . Find the distribution and probability density functions of arctan(Z).

Question 10

Suppose that the amount of water, in thousands of gallons, which fall on a city during a rain storm has distribution function: $$F ( x ) = \left\{ \begin{array} { l l } 1 - \frac { 1 } { e ^ { x } } & x \geq 0 \\0 & \text { elsewhere }\end{array} \right.$$ Find the probability that, in a rain storm, at least 2000 gallons of water fall, given that 1200 gallons have already fallen. Is this different than the probability that 800 gallons fall without any prior knowledge?

Question 11

Assume that a certain dog sled team has race times, in hours, which are a random variable with probability density function $$f ( x ) = \left\{ \begin{array} { l l } \frac { 4 } { 567 } x ^ { 3 } - x & 2 < x < 5 \\0 & \text { else }\end{array} \right.$$
(a) Find the expected duration of such a race.
(b) Find the probability that if the dogs run 5 races, then none of them is shorter than 3 hours. Assume the duration of the individual races are independent of one another.

Question 12

Let X be uniformly distributed on . Find the probability density function and distribution function of $Y = e ^ { X ^ { 2 } }$ .

Question 13

On a certain stretch of highway, to be pulled over by police, the speed at which a car must travel over the speed limit of 60mph is a random variable which has the following probability density function: $$f ( x ) = \left\{ \begin{array} { l l } \frac { 2 x ^ { 2 } - 2 x } { 219 } & 5 < x < 8 \\0 & \text { else }\end{array} \right.$$
(a) Find the maximum speed at which a vehicle can go and be 90% sure that it will not be pulled over.
(b) Suppose Mary-Sue gets pulled over on this highway. What was the expected speed she was driving?

Question 14

Let U be a continuous random variable with probability density function $$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { ( 1 - x ) ^ { 2 } } & 0 \leq x \leq .5 \\0 & \text { else. }\end{array} \right.$$ Using the method of transformations, find the probability density function of $X = e ^ { U + 1 }$ .

Question 15

Suppose that all students of a class of 12 get tested for fever. Furthermore, suppose that the temperature of a student is a random variable with probability density function $$f ( x ) = \left\{ \begin{array} { l l } \frac { 630 } { x ^ { 2 } } & 90 < x < 105 \\0 & \text { elsewhere }\end{array} \right.$$ If 7 or more students have a fever above , the class is dismissed. If the students' temperatures are independent of one another, find the probability that they are dismissed.