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Question 1 of 40 Quiz ID: q1

The time between customer arrivals at a coffee shop follows an exponential distribution with λ = 0.2 customers per minute. What is the probability that the next customer will arrive within 3 minutes?

0.4512

0.5488

0.3297

0.6703

Question 2 of 40 Quiz ID: q2

A light bulb manufacturer claims their bulbs have an average lifespan of 800 hours with exponential distribution. What is the probability that a bulb will last more than 1000 hours?

0.2865

0.7135

0.3679

0.6321

Question 3 of 40 Quiz ID: q3

The time between earthquakes in a certain region follows an exponential distribution with mean 50 years. What is the probability that the next earthquake occurs within the next 30 years?

0.4512

0.5488

0.3297

0.6703

Question 4 of 40 Quiz ID: q4

A server processes requests with exponential service time averaging 2 minutes per request. What is the probability that a request takes more than 3 minutes to process?

0.2231

0.7769

0.3679

0.6321

Question 5 of 40 Quiz ID: q5

The memoryless property of exponential distribution means that if a component has survived 1000 hours, the probability it survives another 500 hours is the same as:

P(surviving 1500 hours)

P(surviving 500 hours)

P(surviving 1000 hours)

P(surviving 2000 hours)

Question 6 of 40 Quiz ID: q6

A radioactive element decays exponentially with half-life of 10 years. What is the decay rate parameter λ?

0.0693 per year

0.1000 per year

0.0500 per year

0.0333 per year

Question 7 of 40 Quiz ID: q7

The time between customer complaints follows exponential distribution with mean 20 days. What is the probability that the next complaint occurs between 10 and 30 days from now?

0.3834

0.6166

0.2331

0.7669

Question 8 of 40 Quiz ID: q8

If X ~ Exp(λ), what is the 90th percentile of the distribution?

-ln(0.1)/λ

-ln(0.9)/λ

ln(0.9)/λ

ln(0.1)/λ

Question 9 of 40 Quiz ID: q9

A call center receives calls exponentially with mean time between calls of 5 minutes. What is the probability that no calls are received in a 10-minute period?

0.1353

0.8647

0.3679

0.6321

Question 10 of 40 Quiz ID: q10

The variance of an exponential distribution with parameter λ = 0.25 is:

16

4

0.0625

0.25

Question 11 of 40 Quiz ID: q11

A gamma distribution with α = 1 and β = 1/λ is equivalent to:

Exponential distribution with parameter λ

Normal distribution with mean λ

Poisson distribution with parameter λ

Binomial distribution with trials λ

Question 12 of 40 Quiz ID: q12

For a gamma distribution with α = 3 and β = 2, what is the mean?

6

12

1.5

5

Question 13 of 40 Quiz ID: q13

For a gamma distribution with α = 3 and β = 2, what is the variance?

12

6

18

24

Question 14 of 40 Quiz ID: q14

The time to complete a complex task follows a gamma distribution with α = 4 and β = 0.5 hours. What is the expected completion time?

2 hours

1 hour

4 hours

0.5 hours

Question 15 of 40 Quiz ID: q15

If X ~ Gamma(α, β), what happens to the shape of the distribution as α increases?

It becomes more symmetric and bell-shaped

It becomes more right-skewed

It becomes more left-skewed

It becomes uniform

Question 16 of 40 Quiz ID: q16

The sum of n independent exponential random variables with parameter λ follows:

Gamma distribution with α = n, β = 1/λ

Exponential distribution with parameter nλ

Normal distribution with mean n/λ

Poisson distribution with parameter nλ

Question 17 of 40 Quiz ID: q17

A gamma distribution is used to model the time until the 5th customer arrival when arrivals are Poisson with rate 2 per hour. What are the parameters α and β?

α = 5, β = 0.5

α = 2, β = 5

α = 5, β = 2

α = 0.5, β = 5

Question 18 of 40 Quiz ID: q18

The gamma function Γ(5) equals:

24

120

6

720

Question 19 of 40 Quiz ID: q19

Γ(1/2) equals:

√π

π

1

0.5

Question 20 of 40 Quiz ID: q20

For a gamma distribution with α = 2.5 and β = 4, what is the mean?

10

6.5

40

12.5

Question 21 of 40 Quiz ID: q21

The time between system failures follows exponential distribution with mean 100 hours. What is the probability that the system fails within 50 hours?

0.3935

0.6065

0.3167

0.6833

Question 22 of 40 Quiz ID: q22

If P(X > 20) = 0.3679 for an exponential distribution, what is the value of λ?

0.05

0.10

0.15

0.20

Question 23 of 40 Quiz ID: q23

The moment generating function of exponential distribution with parameter λ is:

λ/(λ-t) for t < λ

e^(λt) for all t

1/(1-λt) for t < 1/λ

λe^(λt) for t > 0

Question 24 of 40 Quiz ID: q24

A gamma distribution with α = 1 is equivalent to:

Exponential distribution

Normal distribution

Uniform distribution

Poisson distribution

Question 25 of 40 Quiz ID: q25

The time to process insurance claims follows gamma distribution with α = 3 and β = 2 days. What is the probability that a claim takes more than 8 days to process?

0.2381

0.4232

0.5768

0.7619

Question 26 of 40 Quiz ID: q26

The memoryless property is unique to:

Exponential and geometric distributions

Normal and binomial distributions

Gamma and Poisson distributions

Uniform and beta distributions

Question 27 of 40 Quiz ID: q27

If the time between earthquakes follows exponential distribution with mean 30 years, what is the probability that the next earthquake occurs after 40 years but before 60 years?

0.1481

0.8519

0.2636

0.7364

Question 28 of 40 Quiz ID: q28

A gamma distribution with α = 2.5 and β = 4 has variance:

40

10

100

25

Question 29 of 40 Quiz ID: q29

The time between customer arrivals follows exponential distribution. If the probability of no arrivals in 10 minutes is 0.3679, what is the average time between arrivals?

10 minutes

5 minutes

20 minutes

15 minutes

Question 30 of 40 Quiz ID: q30

For a gamma distribution, when α is a positive integer, the CDF can be related to:

Poisson distribution probabilities

Normal distribution probabilities

Binomial distribution probabilities

Uniform distribution probabilities

Question 31 of 40 Quiz ID: q31

A system has components with exponentially distributed lifetimes (mean = 100 hours). What is the probability that at least one of two independent components fails within 50 hours?

0.6321

0.3679

0.8647

0.1353

Question 32 of 40 Quiz ID: q32

The gamma distribution is often used to model:

Time until kth event in Poisson process

Number of events in fixed time interval

Proportion of successes in trials

Measurement errors

Question 33 of 40 Quiz ID: q33

If X ~ Exp(λ), then Y = kX (for k > 0) follows:

Exp(λ/k)

Exp(kλ)

Gamma(1, k/λ)

Normal(k/λ, k²/λ²)

Question 34 of 40 Quiz ID: q34

The median of exponential distribution with parameter λ is:

ln(2)/λ

1/λ

λln(2)

2/λ

Question 35 of 40 Quiz ID: q35

For gamma distribution with α = 4 and β = 3, the mode is:

9

12

6

3

Question 36 of 40 Quiz ID: q36

The skewness of exponential distribution is:

2

1

0

3

Question 37 of 40 Quiz ID: q37

If X ~ Gamma(α, β), then kX (for k > 0) follows:

Gamma(α, kβ)

Gamma(kα, β)

Gamma(α, β/k)

Gamma(kα, kβ)

Question 38 of 40 Quiz ID: q38

The probability density function of gamma distribution includes the term x^(α-1). This means the distribution has mode at 0 when:

α ≤ 1

α ≥ 1

α = 1

α > 2

Question 39 of 40 Quiz ID: q39

The relationship between exponential and Poisson distributions is that:

Time between Poisson events is exponential

Number of exponential events in time is Poisson

Both are memoryless distributions

All of the above

Question 40 of 40 Quiz ID: q40

The gamma function Γ(n) for positive integer n is equivalent to:

(n-1)!

n!

n(n-1)!

√π when n=1/2

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