Question 1

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 probability that more than 25 minutes will pass is given by the survival function of the exponential distribution: P(X > x) = e^(-x/θ). Here, x = 25 and θ = 10, so P(X > 25) = e^(-25/10) = e^(-2.5) ≈ 0.082085.

Question 2

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 probability that the next customer arrives within 5 minutes is given by the cumulative distribution function (CDF) of the exponential distribution: P(X ≤ 5) = 1 - e^(-5/8.5) ≈ 0.444694.

Question 3

Suppose that the random variable x has an exponential distribution with θ = 1.5. Find the probability that x will assume a value within the interval μ ± 2σ.

Accepted Answer

The probability that x will assume a value within the interval μ ± 2σ is given by P(μ - 2σ < x < μ + 2σ) = P(-1 < x < 5) = 0.950213.

Question 4

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

Accepted Answer

For an exponential distribution with parameter θ, the mean (μ) is θ and the standard deviation (σ) is also θ. Therefore, both the mean and standard deviation are 1.5.

Question 5

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...

Question 6

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

Accepted Answer

The cumulative distribution function (CDF) for an exponential distribution is $ P(x < a) = 1 - e^{-a/\theta} $. Substituting $ a = 1.5 $ and $ \theta = 2 $, we get $ P(x < 1.5) = 1 - e^{-1.5/2} = 1 - e^{-0.75} \approx 0.527633 $.

Question 7

Suppose that x has an exponential distribution with $\theta = 2.5 . \text { Find } P ( x \geq 4 )$

Accepted Answer

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

Question 8

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

Accepted Answer

The cumulative distribution function (CDF) for an exponential distribution is $1 - e^{-x/\theta}$. Substituting $x = 10$ and $\theta = 5$, we get $1 - e^{-10/5} = 1 - e^{-2} \approx 0.864665$.

Question 9

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

The answer of The time between arrivals at an ATM...

Question 10

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 probability that the time until the first critical-part failure is 5 years or more is given by the cumulative distribution function of the exponential distribution:$$P(X \ge 5) = 1 - P(X < 5) = 1 - (1 - e^(-5/3.4)) = e^(-5/3.4) = 0.229790$$

Question 11

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

Accepted Answer

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

Question 12

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 μ. b. Find the probability that x assumes a value less than one standard deviation from μ. c. Find the probability that x assumes a value within a half standard deviation of μ.

Accepted Answer

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

Question 13

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 with parameter θ = 10 minutes has a cumulative distribution function (CDF) given by F(x) = 1 - e^(-x/θ). For x = 25, F(25) = 1 - e^(-25/10) = 1 - e^(-2.5) ≈ 0.917915.

Question 14

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 exponential distribution with parameter θ = 10 has a probability density function f(x) = (1/θ) * e^(-x/θ). The probability that the time between arrivals is between 15 and 25 minutes is given by the integral of the density function from 15 to 25, which is F(25) - F(15) = (1 - e^(-25/10)) - (1 - e^(-15/10)) = e^(-15/10) - e^(-25/10) = 0.141045.

Question 15

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 mean of an exponential distribution is the inverse of the rate parameter, λ. Here, λ = 1/10. The probability of waiting more than 25 minutes is given by P(X > 25) = e^(-λ*25) = e^(-25/10) = e^(-2.5) ≈ 0.082085.

Question 16

An online retailer reimburses a customer's 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 customer's shipping...

Question 17

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)...

Question 18

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 with a mean of 3.4 years has a rate parameter λ = 1/3.4. The probability that the time until failure is less than 1 year is given by P(T < 1) = 1 - e^(-λ*1) = 1 - e^(-1/3.4) ≈ 0.254811.

Question 19

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 probability that the next customer will not arrive for at least 20 minutes is given by the exponential distribution's survival function: P(X > 20) = e^(-20/8.5) ≈ 0.095089.

Quiz 5: Continuous Random Variables