Question 1

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.

Accepted Answer

A)  0.64
B)  0.71
C)  0.35
D)  0.78  B

Question 2

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

Accepted Answer

A) True 
 B)False  False

Question 3

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.

Accepted Answer

A)    e4−1e ^ { 4 } - 1e4−1 
B)    e−4+1e ^ { - 4 } + 1e−4+1 
C)    e−4e ^ { - 4 }e−4 
D)    e4e ^ { 4 }e4 C

Question 4

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

Accepted Answer

A) True 
 B)False

Question 5

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

Accepted Answer

A)  3.26
B)  2.18
C)  1.09
D)  2.72

Question 6

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.

Accepted Answer

A)  0.097
B)  0.049
C)  0.194
D)  0.146

Question 7

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.

Accepted Answer

A)  6 minutes
B)  2.5 minutes
C)  3 minutes
D)  1.5 minutes

Question 8

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.

Accepted Answer

A)  0.1901
B)  0.3802
C)  0.2851
D)  0.0950

Question 9

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 $E ( X )$

Accepted Answer

A)  4
B)  0
C)  2
D)  8

Question 10

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?

Accepted Answer

A)  $5.15
B)  $0.92
C)  $9.20
D)  $1.84

Question 11

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

Accepted Answer

A)  0.0166
B)  0.0268
C)  0.0336
D)  0.0212

Question 12

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.

Accepted Answer

A)  1.2
B)  1.1
C)  0.9
D)  1

Question 13

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

Accepted Answer

A) True 
 B)False

Question 14

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 $P ( X > 3 )$

Accepted Answer

A)    516\frac { 5 } { 16 }165​ 
B)    78\frac { 7 } { 8 }87​ 
C)    316\frac { 3 } { 16 }163​ 
D)    1316\frac { 13 } { 16 }1613​

Question 15

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.

Accepted Answer

A)  0.17
B)  0.83
C)  0.62
D)  0.42

Question 16

$f ( x ) = \left\{ \begin{array} { r r } 
\frac { 1 } { 6 } & \text { for } 0 \leq x \leq 6 \
0 & \text { otherwise }
\end{array} \right.$ 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.

Accepted Answer

A)  0.83
B)  0.11
C)  0.06
D)  0.17

Question 17

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?

Accepted Answer

A)  {0; 1; 2}
B)  {1; 2; 3}
C)  {1; 2}
D)  {0; 1; 2; 3}

Question 18

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 $E ( X ) \text { and } E ( Y )$

Accepted Answer

A)    E(X) =6;E(Y) =1E ( X )  = 6 ; E ( Y )  = 1E(X) =6;E(Y) =1 
B)    E(X) =1;E(Y) =5E ( X )  = 1 ; E ( Y )  = 5E(X) =1;E(Y) =5 
C)    E(X) =1;E(Y) =6E ( X )  = 1 ; E ( Y )  = 6E(X) =1;E(Y) =6 
D)    E(X) =7;E(Y) =1E ( X )  = 7 ; E ( Y )  = 1E(X) =7;E(Y) =1

Question 19

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.

Accepted Answer

A)  0.10
B)  0.83
C)  0.17
D)  0.14

Exam 10: Probability and Calculus

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

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.

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 $P ( X > 3 )$

$f ( x ) = \left\{ \begin{array} { r r } \frac { 1 } { 6 } & \text { for } 0 \leq x \leq 6 \\0 & \text { otherwise }\end{array} \right.$ 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?

Exam 10: Probability and Calculus

f(x)={4e−4x for x≥00 for x<0f ( x ) = \left\{ \begin{array} { l l } 4 e ^ { - 4 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.f(x)={4e−4x0​ for x≥0 for x<0​ is a probability density function.

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

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.

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) =142πe−x2/32f ( x ) = \frac { 1 } { 4 \sqrt { 2 \pi } } e ^ { - x ^ { 2 } / 32 }f(x) =42π​1​e−x2/32 Find the expected value E(X) E ( X ) 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≤−2.03) P ( Z \leq - 2.03 ) P(Z≤−2.03) Round your answer to four decimal places.

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

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

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?

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

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

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 $E ( X )$

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 $P ( X > 3 )$