Question

The working life (in years) of a certain type of machines is an exponential random variable with parameter λ, which depends on the quality of its chip. Suppose that the quality of chips are random in a sense that λ is uniformly distributed between [0.5, 1].

1.Find the expected working life of a machine.

2.Find the variance of the working time of a machine.

Answer #1

Given an exponential random variable X with parameter θ and θ is
uniformly distributed on the interval (2,4). Solve for the expected
value and variance of X.

Let X be an exponential random variable with parameter λ > 0.
Find the probabilities P( X > 2/ λ ) and P(| X − 1 /λ | < 2/
λ) .

Let X be a random variable with an exponential
distribution and suppose P(X > 1.5) = .0123
What is the value of λ?
What are the expected value and variance?
What is P(X < 1)?

Suppose that a certain system contains three components that
function independently of each other and are connected in series,
so that the system fails as soon as one of the components fails.
Suppose that the length of life of the first component, X1,
measured in hours, has an exponential distribution with parameter λ
= 0.01; the length of life of the second component, X2, has an
exponential distribution with parameter λ = 0.03; and the length of
life of the...

The battery life of a certain fire alarm, denoted by X, follows
an exponential distribution with mean 3 hours. Suppose we take a
random sample of 36 fire alarms. Answer the following
questions.
A. Find the probability that a single fire alarm will operate
for at least 6 hours.
B. Find the probability that the total battery lifetime based on
the 36 fire alarms will be more than 126 hours.
C.
If the sample mean was 3.5, find a 95%...

Suppose a chandelier has 2019 bulbs, each has independent random
lifetimes that is uniformly distributed on an interval [0,β] with
the same variance as an exp(1/4) variable. (You have to figureout
the value of β.) Find the expected time until 2001 bulbs have
burned out.

The lifespan of a virus depends on the type of RNA it carries.
There are two types of RNA that a virus can randomly carry. The
probability that a virus carries type 1 RNA is 0.4 and the
probability it carries type 2 RNA is 0.6. For a given type of RNA,
the lifespan of a virus is random and can be characterized as
exponential. The average amount of time for the virus to exhibit no
bio-activity is about 4...

Assume that X is a random variable describing the outcomes of a
radar gun used by a police officer to catch drivers exceeding the
speed limit. The radar gun does not record the true speed of the
car. It either records the speed of the driver as 5 miles per hour
(mph) too fast or 5 mph too slow. Suppose the police officer takes
a sample equal to 1. The probability distribution for a sample
equal to 1 is as...

Suppose the breaking strength of plastic bags is a Gaussian
random variable. Bags from company 1 have a mean strength of 8
kilograms and a variance of 1 kg2 ; Bags from company 2 have a mean
strength of 9 kilograms and a variance of 0.5 kg2 . Assume we check
the sample mean ?̅ 10 of the breaking strength of 10 bags, and use
?̅ 10 to determine whether a batch of bags comes from company 1
(null hypothesis...

how can i do this?
The lifetime of a particular type of fluorescent lamp is
exponentially distributed with expectation 1.6 years. Let
T be the life of a random fluorescent lamp. Assume that the
lifetimes of different fluorescent lamps are independent.
a) Show that P (T> 1) = 0.535.
Find P (T <1.6).
In a room, 8 fluorescent lamps of the type are installed. Find
the probability that
at least 6 of these fluorescent lamps still work after one
year....

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