Question

After having misunderstood how long ittook between the passing of cars on a motorway ited the data indicate that these periods are distivited according to the density function F defined by B) (H)=0,25e-⁰.¹³⁵ᵗ (t≥0) where t is the elapsed time in sec

a) What is the probebility that it will take 5 secands or less for him to drive on thet highway?

Answer 1

The probability that it will take 5 seconds or less for the person to drive on the highway is 0.62.

Step-by-step explanation:

To find the probability that it takes 5 seconds or less for the person to drive on the highway, we need to integrate the density function F defined by B) (H) = 0.25e^{-0.135^7} from 0 to 5 seconds.

Let x be the elapsed time in seconds. Then, the probability that it takes x seconds or less for the person to drive on the highway is given by:

P(X ≤ x) = ∫^x_0 F(t) dt

Substituting the given density function, we have:

P(X ≤ x) = ∫^x_0 0.25e^{-0.135^7} dt

Integrating this expression, we get:

P(X ≤ x) = -0.25e^{-0.135^7} |^x_0

P(X ≤ 5) = -0.25e^{-0.135^7} |^5_0

Simplifying this expression, we get:

P(X ≤ 5) = 1 - e^{-0.135^7} = 0.62

Therefore, the probability that it takes 5 seconds or less for the person to drive on the highway is 0.62. This means that there is a relatively high probability that it will take less than 5 seconds for the person to drive on the highway, indicating that there is not much congestion on the highway during this time period.