Chapter 4: Q136E (page 276)

4.135 Suppose xhas an exponential distribution with $θ = 2.5$ . Find
the following probabilities:
$a . P (x \leq 4) b . P (x > 5) c . P (x \leq 2) d . P (x > 3)$

Short Answer

Expert verified

a. The probability is 0.7981

b. The probability is 0.1353

c. The probability is 0.5507

d. The probability is 0.3012

Step by step solution

Given Information

The random variable x has an exponential distribution with $θ = 2.5$

The probability density function of x

Here x is a random variable with parameters $θ = 2.5$

The pdf of x is given by,

$f (x; θ) = \frac{1}{θ} e x p (- \frac{x}{θ}); x > 0$

Here,

$f (x) = \frac{1}{2.5} e x p (- \frac{x}{2.5}); x > 0$

Finding cdf of x

$F (x) = P (X \leq x) = \int_{0}^{x} f (t) d t = \int_{0}^{x} \frac{1}{2.5} e x p (- \frac{t}{2.5}) d t = \frac{1}{2.5} {[\frac{e x p (- \frac{t}{2.5})}{\frac{1}{2.5}}]}_{0}^{x} = e x p (- \frac{x}{2.5}) + 1 = 1 - e x p (- \frac{x}{2.5}) = 1 - e x p (- 0.4 x) F (x) = 1 - e x p (- 0.4 x)$

Finding the probability when P(x≤4)

$P (x > 4) = F (4) = 1 - e x p (- 0.4 \times 4) = 1 - e x p - (- 1.6) = 1 - 0.201897 = 0.798103 ~ 0.7981$

Thus, the required probability is 0.7981.

Finding the probability when (x>5)

$P (x > 5) = 1 - P (x \leq 5) = 1 - F (5) = 1 - [1 - e x p (- 0.4 \times 5)] = e x p (- 2) = 0.135335 ~ 0.1353$

Thus, the required probability is 0.1353.

Finding the probability when P(x≤2)

$P (x \leq 2) = F (2) = 1 - e x p (- 0.4 \times 2) = 1 - e x p (- 0.8) = 1 - 0.449329 = 0.550671 ~ 0.5507$

Thus, the required probability is 0.5507.

Finding the probability when (x>3)

$P (x > 3) = 1 - P (x \leq 3) = 1 - F (3) = 1 - [1 - e x p (- 0.4 \times 3)] = e x p (- 1.2) = 0.301194 ~ 0.3012$

Thus, the required probability is 0.3012.

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4.135 Suppose xhas an exponential distribution with $θ = 2.5$ . Find
the following probabilities:
$a . P (x \leq 4) b . P (x > 5) c . P (x \leq 2) d . P (x > 3)$

Short Answer

Step by step solution

Given Information

The probability density function of x

Finding cdf of x

Finding the probability when P(x≤4)

Finding the probability when (x>5)

Finding the probability when P(x≤2)

Finding the probability when (x>3)

One App. One Place for Learning.

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4.135 Suppose xhas an exponential distribution withθ=2.5 . Findthe following probabilities:a.P(x≤4)b.P(x>5)c.P(x≤2)d.P(x>3)

Short Answer

Step by step solution

Given Information

The probability density function of x

Finding cdf of x

Finding the probability when P(x≤4)

Finding the probability when (x>5)

Finding the probability when P(x≤2)

Finding the probability when (x>3)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Applied Mathematics

Mechanics Maths

Probability and Statistics

Geometry

Study anywhere. Anytime. Across all devices.

4.135 Suppose xhas an exponential distribution with $θ = 2.5$ . Find
the following probabilities:
$a . P (x \leq 4) b . P (x > 5) c . P (x \leq 2) d . P (x > 3)$