# Probability and Statistics Questions and Answers – Weibull Distribution

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This set of Probability and Statistics Multiple Choice Questions & Answers (MCQs) focuses on “Weibull Distribution”.

1. How many parameters are there in Weibull distribution?
a) 1
b) 2
c) 3
d) 4

Explanation: There are 3 parameters in Weibull distribution β is the shape parameter also known as the Weibull slope, η is the scale parameter, γ is the location parameter.

2. Weibull distribution gives the failure rate proportional to the power of time.
a) True
b) False

Explanation: The Weibull distribution function is used to get the failure rate of device proportional to the power of time. We can predict the failure of a device using Weibull distribution.

3. In Weibull distribution, if the value of β equal us equal to one that indicates the failure rate is constant over time.
a) True
b) False

Explanation: In Weibull distribution for β < 1 failure rate that decreases with time. β = 1 have a constant failure rate. β > 1 have a failure rate that increases with time.

4. For β greater than 1 there is an inflection point for Weibull function at ___________
a) (e1/β-1)/e1/β-1
b) (e1/β-1)/e1/β
c) (e1/β-1)/e1/β-2
d) (e1/β-1)/e1/β-5

Explanation: There is an inflection point for Weibull function at (e1/β-1)/e1/β. The function is first convex, then concave after the inflection point.

5. How can we increase the height of the graph of Weibull distribution?
a) if η is decreased while β and γ are constant
b) if η is increased while β and γ are constant
c) if η is constant while β and γ are increased
d) if η is constant while β and γ are decreased

Explanation: If η is decreased while β and γ are kept constant, the distribution is pushed to its left and thus its height increases.

6. What is the effect of the threshold parameter γ on the graph of Weibull distribution?
a) graph shift to left or right
b) graph shifts up or down
c) height of the graph decreases
d) height of the graph increases

Explanation: γ is called a location parameter. The value of γ decides the least time of failure of the device. More the γ more is the graph shifted towards right.

7. What is the mean time to failure if time to failure of a gadget follows Weibull distribution with scale = 1000 hours and shape = 0.5?
a) 2500 hours
b) 1500 hours
c) 3000 hours
d) 2000 hours

Explanation: The failure of the electric bulb follows a Weibull Distribution,
Mean time to failure is given by
1000 × Γ (1+1/0.5)
1000 × 2 = 2000 hours.

8. The mean time of failure is given by the equation __________
a) E(X) = Γ(1/ 2β + 1)
b) E(X) = Γ(1/ β + 1)
c) E(X) = Γ(1/ β + 2)
d) E(X) = Γ(2/ β + 1)

Explanation: The mean time of failure is given by the equation E(X) = Γ(1/ β + 1). This represents the time expected for a device to fail.

9. The time to failure of an electric bulb follows a Weibull distribution with scale = 2000 hours and shape = 0.5. What is the probability that the electric bulb will last more than 4000 hours? What is the mean time to failure?
a) 25.3%
b) 24.3%
c) 26.3%
d) 27.3%

Explanation: The failure of the electric bulb follows a Weibull Distribution
The probability that the bulb will last no more than 3000 hours = WEIBULL (3000, 0.75, 1000, True) = 0.757
Probability that the bulb will last more than 4000 hours = 1 – 0.757 = 24.3%.

10. γ in Weibull distribution graph indicates _________
a) earliest time of failure
b) maximum time of failure
c) shape of the graph
d) scale of the graph

Explanation: γ in Weibull distribution graph indicates the location for the graph of a density function. The starting point gives the least time of a device to fail.

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