The time till failure of a particular TV brand has an exponential distribution. Data shows that 28% of TV'S have failed by 1500 hours. What is probability that TV's will still be working after 7600 hours

To solve this problem, we'll utilize the properties of the **exponential distribution**.

The exponential distribution's probability density function (PDF) is given by:

\[
f(t) = \lambda e^{-\lambda t}
\]

Where:
- \( \lambda \) is the rate parameter (failure rate).
- \( t \) is the time.

The cumulative distribution function (CDF), which gives the probability that the TV has failed by time \( t \), is:

\[
F(t) = 1 - e^{-\lambda t}
\]

Given that 28% of the TVs have failed by 1500 hours, we can use this information to first find the rate parameter \( \lambda \).

### Step 1: Find the rate parameter \( \lambda \)
We know that:

\[
F(1500) = 0.28
\]

Substitute into the CDF equation:

\[
0.28 = 1 - e^{-\lambda \cdot 1500}
\]

This simplifies to:

\[
e^{-\lambda \cdot 1500} = 0.72
\]

Now, take the natural logarithm of both sides:

\[
-\lambda \cdot 1500 = \ln(0.72)
\]

Solve for \( \lambda \):

\[
\lambda = -\frac{\ln(0.72)}{1500}
\]

### Step 2: Find the probability the TV will still be working after 7600 hours
The probability that the TV is still working after 7600 hours is the complement of the probability of failure by 7600 hours. This is given by:

\[
P(T > 7600) = 1 - F(7600) = e^{-\lambda \cdot 7600}
\]

We now need to compute \( \lambda \), then use it to find this probability. Let me calculate that.The rate parameter \( \lambda \) is approximately \( 0.000219 \).

The probability that the TV will still be working after 7600 hours is approximately **0.189**, or about **18.93%**.

Would you like more details or have any further questions?

Here are five related questions for further exploration:
1. How is the exponential distribution applied in reliability engineering?
2. What is the relationship between the mean lifetime and the rate parameter \( \lambda \)?
3. How do we compute the median lifetime for an exponential distribution?
4. How would the probability change if a different percentage of TVs failed by 1500 hours?
5. How does the exponential distribution compare to the normal distribution in terms of modeling failure times?

**Tip:** In an exponential distribution, the expected time to failure is the reciprocal of the rate parameter, \( E(T) = \frac{1}{\lambda} \).

The time till failure of a particular TV brand has an exponential distribution. Data shows that 28% of TV'S have failed by 1500 hours. What is the probability that TV's will still be working after 7600 hours?

Explore how to calculate the probability that a TV is still functioning after 7600 hours using exponential distribution. With 28% of TVs failing by 1500 hours, learn step-by-step how to determine the failure rate and solve the problem, ideal for students in advanced statistics courses.

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