Question : The length of time(in hundreds of hours) for the failure of a transistor is a random variable Y with distribution function ;
F(Y) = { 0 : if y<0 or [1 - e^(-(y^2) ] if y>=0 }
Find the probability that the transistor operates for at least 200 hours.
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Original Post:
The length of time(in hundreds of hours) for the failure of a transistor is a random variable Y with distribution function ;
F(Y) = { 0 : if y<0 or [1 - e^(-(y^2) ] if y>=0 }
Find the probability that the transistor operates for at least 200 hours.
Subject: Statistics
Current Topic: Cumulative Probability Distribution
Original Post
The length of time(in hundreds of hours) for the failure of a transistor is a random variable Y with distribution function ;
F(Y) = { 0 : if y<0 or [1 - e^(-(y^2) ] if y>=0 }
Find the probability that the transistor operates for at least 200 hours.
Since we need to find the probability that the transistor operates for at least 200 hours, we nedd to find, P(Y>=200) = integrate [f(t) dt] from -infinity to 200.
P(Y>=200) = integrate [ 0 dt] from -infinity to 0 + integrate [1 - (e^(-(t^2))) dt] from 0 to 200
==> P(Y>=200) = 0 + integrate [1 - (e^(-(t^2))) dt] from 0 to 200
==> P(Y>=200) = integrate [1 - (e^(-(t^2))) dt] from 0 to 200
==> P(Y>=200) = integrate [1 dt ] from 0 to 200 + integrate [ - (e^(-(t^2))) dt] from 0 to 200
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Original Post:
The length of time(in hundreds of hours) for the failure of a transistor is a random variable Y with distribution function ;
F(Y) = { 0 : if y<0 or [1 - e^(-(y^2) ] if y>=0 }
Find the probability that the transistor operates for at least 200 hours.
Subject: Statistics
Current Topic: Cumulative Probability Distribution
Original Post
The length of time(in hundreds of hours) for the failure of a transistor is a random variable Y with distribution function ;
F(Y) = { 0 : if y<0 or [1 - e^(-(y^2) ] if y>=0 }
Find the probability that the transistor operates for at least 200 hours.
But, how do we the following integration? I'm having trouble finding a suitable substitution or any other method for the following integration ; I = integrate [1 - (e^(-(t^2))) dt] from 0 to 200
Or is there any other method we can use to do this problem?
Thank you!
To solve the integral I = integrate [1 - (e^(-(t^2))) dt] from 0 to 200, there is no simple closed form solution. However, it can be approximated using numerical methods.
One common method is to use numerical integration techniques, such as the trapezoidal rule or Simpson's rule. These methods approximate the integral by dividing the interval into smaller segments and approximating each segment using a polynomial. The accuracy of the approximation depends on the number of segments used.
Alternatively, you can use software or programming language with built-in numerical integration functions, such as MATLAB, Python with NumPy, or R. These tools provide functions that can directly compute the integral numerically.
Here is an example of how to solve the integral using Python with NumPy:
```python
import numpy as np
from scipy.integrate import quad
def integrand(t):
return 1 - np.exp(-t**2)
result, error = quad(integrand, 0, 200)
print(result)
```
This code uses the `quad` function from the `scipy.integrate` module in the SciPy library. The `integrand` function defines the integrand, and the `quad` function computes the integral. The result is stored in the `result` variable.
Note that the result may be an approximation due to the numerical nature of the integration methods. Additionally, the computation time may vary depending on the complexity and accuracy desired.
I hope this helps! Let me know if you have any further questions.