Question 1

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.

Accepted Answer

The probability that the time between arrivals is between 15 and 25 minutes is given by the difference of the cumulative distribution function (CDF) of the exponential distribution: P(15 < X < 25) = e^(-15/10) - e^(-25/10) = 0.141045.

Question 2

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$. Find the mean and standard deviation of $x$.

Accepted Answer

For an exponential distribution with parameter $\theta$, the mean ($\mu$) and standard deviation ($\sigma$) are both equal to $\theta$. Therefore, with $\theta = 1.5$, both the mean and standard deviation are 1.5.

Question 3

An online retailer reimburses a customerʹs shipping charges if the customer does not
receive his order within one week. Delivery time (in days)is exponentially distributed
with a mean of 3.2 days. What percentage of customers have their shipping charges
reimbursed?

Accepted Answer

The answer of An online retailer reimburses a customerʹs shipping...

Question 4

The time between customer arrivals at a furniture store has an approximate exponential distribution with mean θ = 8.5 minutes. If a customer just arrived, find the probability that the next
Customer will not arrive for at least 20 minutes.

Accepted Answer

The exponential distribution is defined by the probability density function f(x|θ) = (1/θ) * e^(-x/θ). The probability that the time until the next event is at least t is given by P(X ≥ t) = e^(-t/θ). Here, θ = 8.5 and t = 20, so P(X ≥ 20) = e^(-20/8.5) ≈ 0.095089.

Question 5

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the mean and standard deviation of this distribution.

Accepted Answer

For an exponential distribution with rate parameter $ \theta $, the mean (or expected value) is $ \theta $ and the standard deviation is also $ \theta $. Therefore, with $ \theta = 10 $, both the mean and standard deviation are 10.

Question 6

The time between equipment failures (in days)at a particular factory is exponentially
distributed with a mean of 4.5 days. A machine just failed and was repaired today.
a. Find the probability that another machine will fail within the next day.
b. Find the probability that there will be no more equipment failures in the next week.

Accepted Answer

The answer of The time between equipment failures (in days)at...

Question 7

Suppose that $x$ has an exponential distribution with $\theta = 5$. Find $P ( x \leq 10 )$.

Accepted Answer

The cumulative distribution function (CDF) for an exponential distribution is given by $1 - e^{-\frac{x}{\theta}}$. Substituting $x = 10$ and $\theta = 5$ gives $1 - e^{-\frac{10}{5}} = 1 - e^{-2} \approx 0.864665$.

Question 8

Suppose that $x$ has an exponential distribution with $\theta = 2$. Find $P ( x < 1.5 )$.

Accepted Answer

The cumulative distribution function (CDF) for an exponential distribution is given by $1 - e^{-\frac{x}{\theta}}$. Substituting $x = 1.5$ and $\theta = 2$ into the CDF formula gives $1 - e^{-\frac{1.5}{2}} = 1 - e^{-0.75} \approx 0.527633$.

Question 9

The time (in years)until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is 5
Years or more.

Accepted Answer

The exponential distribution is given by the formula $P(X > x) = e^{-x/\lambda}$, where $\lambda$ is the mean. Here, $\lambda = 3.4$ years, and we want $P(X > 5)$. Thus, $P(X > 5) = e^{-5/3.4} \approx 0.229790$.

Question 10

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that more than 25 minutes will pass between arrivals.

Accepted Answer

The exponential distribution's probability density function is given by f(x;θ) = (1/θ) * e^(-x/θ). The probability that more than 25 minutes will pass is P(X > 25) = e^(-25/10) = e^(-2.5) ≈ 0.082085.

Question 11

The time (in years)until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is less
Than 1 year.

Accepted Answer

The exponential distribution is given by the formula $P(X < x) = 1 - e^{-\lambda x}$, where $\lambda = \frac{1}{\text{mean}}$. Here, the mean is 3.4 years, so $\lambda = \frac{1}{3.4}$. For $x = 1$ year, $P(X < 1) = 1 - e^{-\frac{1}{3.4}} \approx 0.254811$.

Question 12

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$. Find the probability that $x$ will assume a value within the interval $\mu \pm 2 \sigma$.

Accepted Answer

For an exponential distribution, the mean ($\mu$) and standard deviation ($\sigma$) are both equal to $\theta$. Therefore, $\mu = \sigma = 1.5$. The interval $\mu \pm 2\sigma$ translates to $1.5 \pm 3$, or $-1.5$ to $4.5$, but since negative values are not possible for an exponential distribution, we only consider the upper limit of $4.5$. The probability that $x$ will assume a value within $\mu \pm 2\sigma$ for any normal or approximately normal distribution (and by extension, using the empirical rule for exponential distributions in a broad sense) is approximately 95%. Hence, the closest answer is $.950213$.

Question 13

The waiting time (in minutes)between ordering and receiving your meal at a certain restaurant is exponentially distributed with a mean of 10 minutes. The restaurant has a policy that your meal is
Free if you have to wait more than 25 minutes after ordering. What is the probability of receiving a
Free meal?

Accepted Answer

The waiting time being exponentially distributed with a mean of 10 minutes means the rate parameter $ \lambda = \frac{1}{10} $. The probability of waiting more than 25 minutes is $ P(T > 25) = e^{-\lambda t} = e^{-\frac{25}{10}} = e^{-2.5} \approx 0.082085 $.

Question 14

The length of time (in months)that a cashier works for a certain fast food restaurant is
exponentially distributed with a mean of 7 months.
a. Find the probability that a cashier works for the restaurant for at least 2 years.
b. Find the probability that a cashier works for the restaurant for less than 1 month.

Accepted Answer

The answer of The length of time (in months)that a...

Question 15

Suppose that the random variable x has an exponential distribution with θ = 3. a. Find the probability that $x$ assumes a value more than three standard deviations from $\mu$.
b. Find the probability that $x$ assumes a value less than one standard deviation from $\mu$.
c. Find the probability that $x$ assumes a value within a half standard deviation of $\mu$.

Accepted Answer

The answer of Suppose that the random variable x has...

Question 16

The time between customer arrivals at a furniture store has an approximate exponential distribution with mean θ = 8.5 minutes. If a customer just arrived, find the probability that the next
Customer will arrive in the next 5 minutes.

Accepted Answer

The exponential distribution is given by the formula $P(X \leq x) = 1 - e^{-\frac{x}{\theta}}$, where $\theta$ is the mean. Plugging in $x = 5$ minutes and $\theta = 8.5$ minutes, we get $P(X \leq 5) = 1 - e^{-\frac{5}{8.5}} \approx 0.444694$.

Question 17

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that less than 25 minutes will pass between arrivals.

Accepted Answer

The exponential distribution is given by the formula $P(X < x) = 1 - e^{-\frac{x}{\theta}}$, where $\theta$ is the mean time between arrivals. Plugging in the values, we get $P(X < 25) = 1 - e^{-\frac{25}{10}} = 1 - e^{-2.5} \approx 0.917915$.

Question 18

Suppose that $x$ has an exponential distribution with $\theta = 1.75$. Find each probability.
a. $P ( x \geq 2 )$
b. $P ( x < 2 )$

Accepted Answer

The answer of Suppose that $x$ has an exponential distribution...

Quiz 5: Continuous Random Variables

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$ . Find the mean and standard deviation of $x$ .

An online retailer reimburses a customerʹs shipping charges if the customer does not receive his order within one week. Delivery time (in days)is exponentially distributed with a mean of 3.2 days. What percentage of customers have their shipping charges reimbursed?

The time between customer arrivals at a furniture store has an approximate exponential distribution with mean θ = 8.5 minutes. If a customer just arrived, find the probability that the next Customer will not arrive for at least 20 minutes.

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the mean and standard deviation of this distribution.

Suppose that $x$ has an exponential distribution with $\theta = 5$ . Find $P ( x \leq 10 )$ .

Suppose that $x$ has an exponential distribution with $\theta = 2$ . Find $P ( x < 1.5 )$ .

The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is 5 Years or more.

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that more than 25 minutes will pass between arrivals.

The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is less Than 1 year.

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$ . Find the probability that $x$ will assume a value within the interval $\mu \pm 2 \sigma$ .

The time between customer arrivals at a furniture store has an approximate exponential distribution with mean θ = 8.5 minutes. If a customer just arrived, find the probability that the next Customer will arrive in the next 5 minutes.

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that less than 25 minutes will pass between arrivals.

Suppose that $x$ has an exponential distribution with $\theta = 1.75$ . Find each probability. a. $P ( x \geq 2 )$ b. $P ( x < 2 )$

Quiz 5: Continuous Random Variables

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.

Suppose that the random variable xxx has an exponential distribution with θ=1.5\theta = 1.5θ=1.5 . Find the mean and standard deviation of xxx .

An online retailer reimburses a customerʹs shipping charges if the customer does not receive his order within one week. Delivery time (in days)is exponentially distributed with a mean of 3.2 days. What percentage of customers have their shipping charges reimbursed?

The time between customer arrivals at a furniture store has an approximate exponential distribution with mean θ = 8.5 minutes. If a customer just arrived, find the probability that the next Customer will not arrive for at least 20 minutes.

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the mean and standard deviation of this distribution.

Suppose that xxx has an exponential distribution with θ=5\theta = 5θ=5 . Find P(x≤10) P ( x \leq 10 ) P(x≤10) .

Suppose that xxx has an exponential distribution with θ=2\theta = 2θ=2 . Find P(x<1.5) P ( x < 1.5 ) P(x<1.5) .

The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is 5 Years or more.

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that more than 25 minutes will pass between arrivals.

The time (in years) until the first critical-part failure for a certain car is exponentially distributed with a mean of 3.4 years. Find the probability that the time until the first critical-part failure is less Than 1 year.

Suppose that the random variable xxx has an exponential distribution with θ=1.5\theta = 1.5θ=1.5 . Find the probability that xxx will assume a value within the interval μ±2σ\mu \pm 2 \sigmaμ±2σ .

The time between customer arrivals at a furniture store has an approximate exponential distribution with mean θ = 8.5 minutes. If a customer just arrived, find the probability that the next Customer will arrive in the next 5 minutes.

The time between arrivals at an ATM machine follows an exponential distribution with θ = 10 minutes. Find the probability that less than 25 minutes will pass between arrivals.

Suppose that xxx has an exponential distribution with θ=1.75\theta = 1.75θ=1.75 . Find each probability. a. P(x≥2)P ( x \geq 2 )P(x≥2) b. P(x<2)P ( x < 2 )P(x<2)

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$ . Find the mean and standard deviation of $x$ .

Suppose that $x$ has an exponential distribution with $\theta = 5$ . Find $P ( x \leq 10 )$ .

Suppose that $x$ has an exponential distribution with $\theta = 2$ . Find $P ( x < 1.5 )$ .

Suppose that the random variable $x$ has an exponential distribution with $\theta = 1.5$ . Find the probability that $x$ will assume a value within the interval $\mu \pm 2 \sigma$ .

Suppose that $x$ has an exponential distribution with $\theta = 1.75$ . Find each probability. a. $P ( x \geq 2 )$ b. $P ( x < 2 )$