The time (in minutes) until the next bus departs a major bus depot follows a uniform distribution from 25 to 46 minutes. Let X denote the time until the next bus departs. The distribution is and is. The mean of the distribution is μ=. The standard deviation of the distribution is σ=. The probability that the time until the next bus departs is between 30 and 40 minutes is P (30

Question

Jada Huber

Mathematics

3 March, 18:27

The time (in minutes) until the next bus departs a major bus depot follows a uniform distribution from 25 to 46 minutes. Let X denote the time until the next bus departs. The distribution is and is. The mean of the distribution is μ=. The standard deviation of the distribution is σ=. The probability that the time until the next bus departs is between 30 and 40 minutes is P (30

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Sam Johnston · Answer 1

Step-by-step explanation:

The distribution is Uniform and is continuous.

We are given a = 25 and b = 446

The mean of the distribution is

Mean = µ = (a + b) / 2

= (25 + 44) / 2

= 34.5

The standard deviation of the distribution is

SD = σ = (b - a) / sqrt (12)

= (46 - 25) / sqrt (12)

= 21/3.464102

= 6.062minutes

SD = σ = 6.062minutes

Now, we have to find P (30
P (30
= (40 - 30) / (46 - 25)

= 10/19

= 0.4762

Required probability = 0.4762

Now, we have to find the value of a for which P (X≤a) = 0.9

We have, P (X≤a)

= (a - 25) / (46 - 25)

= (a - 25) / 21

(a - 25) / 21 = 0.9

(a - 25) = 0.9*21

So, a = 18.9 + 25 = 43.

a = 43.9

= 43.9minutes

Sasha Fischer · Answer 2

The distribution is uniform and is continuous probability distribution.

The mean of the distribution is μ = 35.5 minutes.

The standard deviation of the distribution is σ = 6.06

P (30
Step-by-step explanation:

a. The amount of time in minutes until the next bus departs is uniformly distributed between 25 and 46 inclusive.

The distribution is uniform and is continuous probability distribution.

b. Let X be the number of minutes the next bus arrives and a = 25, b = 46. Hence mean = a+b/2 = 25 + 46 / 2 = 35.5 minutes.

The mean of the distribution is μ = 35.5 minutes.

c. Standard deviation = √ (b-a) ²/12

Standard deviation = √ (46-25) ²/12 = √ (21) ²/12 = √441/12=√36.75 = 6.06

The standard deviation of the distribution is σ = 6.06

d. P (30
For this we draw a graph. It is also called the rectangular distribution because its total probability is confined to a rectangular region with base equal to (b-a) and height 1 / (b-a).

This can be calculated as

P (30
= (40-30) * (1 / 21) = 10/21 = 0.476