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MrRissso [65]
2 years ago
15

The number of years a radio functions is exponentially distributed with parameter λ = 1 8 . If Jones buys a used radio, what is

the probability that it will be working after an additional 8 years?
Mathematics
1 answer:
blsea [12.9K]2 years ago
8 0

Answer:

P(X>8)=e^{-1}

Step-by-step explanation:

The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution". The probability density function is given by:

P(X=x)=\lambda e^{-\lambda x}, x>0

And 0 for other case. Let X the random variable that represent "The number of years a radio functions" and we know that the distribution is given by:

X \sim Exp(\lambda=\frac{1}{8})

We can assume that the random variable t represent the number of years that the radio is already here. So the interest is find this probability:

P(X>8|X>t)

We have an important property on the exponential distribution called "Memoryless" property and says this:

P(X>a+t| X>t)=P(X>a)

Where a represent a shift and t the time of interest.

On this case then P(X>8|X>t)=P(X>8+t|X>t)=P(X>8)

We can use the definition of the density function and find this probability:

P(X>8)=\int_{8}^{\infty} \frac{1}{8}e^{-\frac{1}{8}x}dx

=\frac{1}{8} \int_{8}^{\infty} e^{-\frac{1}{8}x}dx

=[lim_{x\to\infty} (-e^{-\frac{1}{8}x})+e^{-1}]=0+e^{-1}=e^{-1}

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