Exam 10: Probability and Calculus

The life span of car stereos manufactured by a certain company is measured by a random variable X that is exponentially distributed with a probability density function $f ( x ) = \left\{ \begin{array} { l l } 0.2 e ^ { - 0.2 x } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ where x is the life span in years of a randomly selected stereo.What is the probability that the life span of a randomly selected stereo is between 1 and 11 years? Round to the nearest hundredth.

$f ( x ) = \left\{ \begin{array} { l l } 4 e ^ { - 4 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ is a probability density function.

Suppose the length of time,x,that it takes a chimpanzee to solve a simple puzzle is measured by a random variable X that is exponentially distributed with a probability density function $f ( x ) = \left\{ \begin{array} { l l } \frac { 2 } { 3 } e ^ { - 2 \times B } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ where x is in minutes.Find the probability that a randomly chosen chimpanzee will take more than 6 minutes to solve the puzzle.

If X has a normal distribution with $\mu = 2$ then $P ( X \leq 1.76 ) = P ( X \geq 2.24 )$

$f ( x ) = \left\{ \begin{array} { c l } \frac { 2 x } { 49 } & \text { if } 0 \leq x \leq 7 \\0 & \text { otherwise }\end{array} \right.$ is a probability density function for a continuous random variable X. Find the variance.Round to the nearest hundredth.

Suppose that the number of peanuts in a bite-sized chocolate candy follows a Poisson distribution with a mean of 3.8 peanuts per candy.Find the probability that a candy contains exactly 4 peanuts.Round to three decimal places.

Suppose the time X a customer must spend waiting in line at a certain fast food restaurant is a random variable that is exponentially distributed with a density function $f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 3 } e ^ { - x / 3 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ where x is the number of minutes that a randomly selected customer spends waiting in line.Find the expected waiting time for customers at the restaurant.

The gestation period of humans follows an approximately normal distribution with a mean of 266 days and a standard deviation of 16 days.Find the probability that a gestation period is between 260 and 276 days.Round to four decimal places.

The density function of a normal random variable X is $f ( x ) = \frac { 1 } { 4 \sqrt { 2 \pi } } e ^ { - x ^ { 2 } / 32 }$ Find the expected value $E ( X )$

The grand prize in a lottery is $500,000.There are 6 second prizes of $50,000 and 12 third prizes of $10,000.If 1,000,000 tickets are sold,what is a fair price to pay for a ticket to this lottery?

Assuming that the random variable Z has a standard normal distribution,find $P ( Z \leq - 2.03 )$ Round your answer to four decimal places.

Find the appropriate value of b for $P ( - b \leq Z \leq b ) = 0.6827$ assuming that the random variable Z has a standard normal distribution.

Suppose X has a normal distribution with $\mu = 16 \text { and } \sigma = 3$ $P ( X \leq 19 ) = 0.4207$

Consider the random experiment of tossing a fair coin five times,and let X denote the random variable that counts the number of times heads appears.Find $P ( X > 3 )$

The time interval between the arrivals of successive trains at a certain station is measured by a random variable X with a probability density function $f ( x ) = \left\{ \begin{array} { l l } 0.2 e ^ { - 0.2 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$ where x is the time (in minutes)between the arrivals of a randomly selected pair of successive trains.What is the probability that two successive trains selected at random will arrive within 9 minutes of one another? Round to the nearest hundredth.

$f ( x ) = \left\{ \begin{array} { r r } \frac { 1 } { 6 } & \text { for } 0 \leq x \leq 6 \\0 & \text { otherwise }\end{array} \right.$ is a probability density function for a particular random variable X.Use integration to find $P ( 3 \leq x \leq 4 )$ rounded to the nearest hundredth.

A die is rolled two times and the number of 2s that are rolled is noted.What is the sample space for this random experiment?

The joint probability density function for two continuous random variables X and Y is $f ( x , y ) = \left\{ \begin{array} { l l } \frac { 1 } { 6 } e ^ { - x / 6 } e ^ { - y } & \text { if } x \geq 0 \text { and } y \geq 0 \\0 & \text { otherwise }\end{array} \right.$ Find the expected values $E ( X ) \text { and } E ( Y )$

A 2-hour movie runs continuously at a local theater.You leave for the theater without first checking the show times.Use an appropriate uniform density function to find the probability that you will arrive at the theater within 10 minutes of the start of the film (before or after).Round the the nearest hundredth.