Question 1

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.

A)

e ^ { 4 } - 1

B)

e ^ { - 4 } + 1

C)

e ^ { - 4 }

D)

e ^ { 4 }

e ^ { - 4 }

Accepted Answer

The exponential distribution has the property that the probability a random variable X is greater than a value x is given by $P(X > x) = e^{-\lambda x}$, where $\lambda$ is the rate parameter of the distribution. In the given function, the rate parameter $\lambda$ is $2/3$, but the function is incorrectly written as $f(x) = \frac{2}{3}e^{-2 \times B}$. It should be $f(x) = \frac{2}{3}e^{-\frac{2}{3}x}$ for $x \geq 0$. To find the probability that it takes more than 6 minutes, we use the correct form of the exponential distribution: $P(X > 6) = e^{-\lambda x} = e^{-\frac{2}{3} \times 6} = e^{-4}$.

Question 2

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.

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

0.194

Accepted Answer

The probability of a candy containing exactly 4 peanuts is given by the Poisson probability mass function:

P(X = 4) = (e^-λ * λ^k) / k!
where λ=3.8 and k=4

P(X = 4) = (e^-3.8 * 3.8^4) / 4! 
P(X = 4) = 0.194 (rounded to three decimal places)

Therefore, the answer is C.

Question 3

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.

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

0.17

Accepted Answer

The movie runs continuously, so the start times are uniformly distributed over a 2-hour period (120 minutes). The favorable time window is 20 minutes (10 minutes before and after the start). The probability is 20/120 = 0.17.

Question 4

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?&#10;A){0; 1; 2}&#10;B){1; 2; 3}&#10;C){1; 2}&#10;D){0; 1; 2; 3}

Accepted Answer

The sample space for the number of 2s rolled in two dice rolls includes 0 (no 2s rolled), 1 (a 2 rolled on one of the rolls), and 2 (a 2 rolled on both rolls).

Question 5

$f ( x ) = \left\{ \begin{array} { r r } \frac { 1 } { 6 } & \text { for } 0 \leq x \leq 6 \\0 & \text { otherwise }\end{array} \right.$ 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)0.83
B)0.11
C)0.06
D)0.17

Accepted Answer

The answer of \(f ( x ) = \left\{ \begin{array}...

Question 6

$f ( x ) = \left\{ \begin{array} { l l } &#10;4 e ^ { - 4 x } & \text { for } x \geq 0 \&#10;0 & \text { for } x < 0&#10;\end{array} \right.$ is a probability density function.

Accepted Answer

The answer of \(f ( x ) = \left\{ \begin{array}...

Question 7

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.

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

Accepted Answer

To find the probability that a gestation period is between 260 and 276 days, we use the Z-score formula: $Z = \frac{X - \mu}{\sigma}$, where $X$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation. For $X = 260$, $Z = \frac{260 - 266}{16} = -0.375$. For $X = 276$, $Z = \frac{276 - 266}{16} = 0.625$. Looking these Z-scores up in a standard normal distribution table or using a calculator, we find the probabilities: $P(Z < 0.625) - P(Z < -0.375)$. This difference gives us the probability of a gestation period being between 260 and 276 days, which is approximately 0.3802.

Question 8

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 $P ( X > 3 )$&#10;A) $\frac { 5 } { 16 }$&#10;B) $\frac { 7 } { 8 }$&#10;C) $\frac { 3 } { 16 }$&#10;D) $\frac { 13 } { 16 }$

Accepted Answer

The probability $P(X > 3)$ means we are looking for the probability of getting more than 3 heads in 5 tosses, which includes getting 4 heads or 5 heads. The total number of outcomes is $2^5 = 32$. The number of ways to get exactly 4 heads is $5C4 = 5$, and the number of ways to get exactly 5 heads is $5C5 = 1$. So, the total favorable outcomes are $5 + 1 = 6$. Therefore, $P(X > 3) = \frac{6}{32} = \frac{3}{16}$.

Question 9

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.&#10;A)1.2&#10;B)1.1&#10;C)0.9&#10;D)1

Accepted Answer

For a standard normal distribution, $P(-b \leq Z \leq b) = 0.6827$ corresponds to approximately one standard deviation from the mean, which is $b = 1$.

Question 10

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

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

Accepted Answer

The answer of \(f ( x ) = \left\{ \begin{array}...

Question 11

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?&#10;A)$5.15&#10;B)$0.92&#10;C)$9.20&#10;D)$1.84

Accepted Answer

The answer of The grand prize in a lottery is...

Question 12

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.

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

Accepted Answer

The answer of Suppose the time X a customer must...

Question 13

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 $E ( X )$&#10;A)4&#10;B)0&#10;C)2&#10;D)8

Accepted Answer

The answer of The density function of a normal random...

Question 14

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.

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

Accepted Answer

The answer of The time interval between the arrivals of...

Question 15

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

Accepted Answer

The answer of If X has a normal distribution with...

Question 16

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

Accepted Answer

The answer of Suppose X has a normal distribution with...

Question 17

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.

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

Accepted Answer

The answer of The life span of car stereos manufactured...

Question 18

Assuming that the random variable Z has a standard normal distribution,find $P ( Z \leq - 2.03 )$ Round your answer to four decimal places.&#10;A)0.0166&#10;B)0.0268&#10;C)0.0336&#10;D)0.0212

Accepted Answer

The answer of Assuming that the random variable Z has...

Question 19

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)

E ( X ) = 6 ; E ( Y ) = 1

B)

E ( X ) = 1 ; E ( Y ) = 5

C)

E ( X ) = 1 ; E ( Y ) = 6

D)

E ( X ) = 7 ; E ( Y ) = 1

Accepted Answer

The answer of The joint probability density function for two...

Deck 10: Probability and Calculus