Question

Suppose that X has an exponential distribution with a mean of 10. Determine the following: a. P(X < 5) b. P(X < 15 | X &gt; 10) c. Compare the results in parts (a) and (b) and comment on the role of the lack of memory property.exponential distribution with a mean of 10. Determine the following: a. P(X < 5) b. P(X < 15 | X > 10) c. Compare the results in parts (a) and (b) and comment on the role of the lack of memory property.

Answer 1

Answer:

a) 0.3935

a) 0.3935

Memory less property of exponential distribution

Step-by-step explanation:

We are given that X has a exponential distribution.

Thus, it will have a  cumulative probability distribution of the form

$P(X \leq x) = 1 - e^{-\lambda x}$, where $\frac{1}{\lambda}$ is the mean of distribution.

We are given  $\frac{1}{\lambda} = 10$

Thus,

$P(X \leq x) = 1 - e^{\frac{-x}{10}}$

a)$P( x < 5) = 1 - e^{\frac{-5}{10}} = 0.3935$

b)

$P(x < 15|x > 10) = \displaystyle\frac{P(15 < x < 10)}{P(x > 10)} \\\\= \displaystyle\frac{(1 - e^{\frac{-15}{10} }) - (1 - e^{\frac{-10}{10} }) }{1 - P(x < 10)}\\\\=\displaystyle\frac{(1 - e^{\frac{-15}{10} }) - (1 - e^{\frac{-10}{10} }) }{1 - 1 + e^{\frac{-10}{10} }}\\\\= 0.3935$

The, answers for a) and b) are same due to memory less property of the exponential distribution.

The future outcomes are not affected by past outcomes.

thus,

$P(X \leq x) = P(x < 15|x > 10)$