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Pie
3 years ago
13

An exponential random variable has a standard deviation of 16. Calculate The probability that it takes on a value < 11. Use 3

decimal places.
Mathematics
1 answer:
antiseptic1488 [7]3 years ago
5 0

Answer:

P(X

And using the cdf we got:

P(X

Step-by-step explanation:

Previous concepts

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 random variable of interest and we know that the distribution is given by:

X \sim Exp(\lambda)

We know the variance on this case given by :

Var(X) = \frac{1}{\lambda^2}

So then the deviation is given by:

Sd(X) = \frac{1}{\sqrt{\lambda}}

And if we solve for \lambda we got:

\lambda = (\frac{1}{16})^2 = \frac{1}{256}

The cumulative distribution function for the exponential distribution is given by:

F(x) = 1- e^{-\lambda x}

Solution to the problem

And for this case we want to find this probability:

P(X

And using the cdf we got:

P(X

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