0. The time required to repair a machine is an exponential random variable with rate $$ \lambda=0.5 $$ owns/hour. What is the probability that a repair time exceeds 2 hours? What is the probability that the repait time will take at least 4 hours given that the repair an been working on the machine for 3 hours?

Question

UpStudy AI · Accepted Answer

To solve the problem, we will use the properties of the exponential distribution.

1. **Probability that a repair time exceeds 2 hours:**
   The probability that an exponential random variable $$ T $$ exceeds a certain value $$ t $$ is given by the formula:
   $$
   P(T > t) = e^{-\lambda t}
   $$
   where $$ \lambda $$ is the rate parameter.

Here, $$ \lambda = 0.5 $$ and $$ t = 2 $$.

2. **Probability that the repair time will take at least 4 hours given that the repair has been working on the machine for 3 hours:**
   This is a conditional probability problem. For an exponential distribution, the memoryless property states that:
   $$
   P(T > s + t | T > s) = P(T > t)
   $$
   In this case, we want to find $$ P(T > 4 | T > 3) $$, which simplifies to:
   $$
   P(T > 4 | T > 3) = P(T > 1)
   $$
   where $$ s = 3 $$ and $$ t = 1 $$.

Now, let's calculate both probabilities step by step.

### Step 1: Calculate $$ P(T > 2) $$
Using the formula:
$$
P(T > 2) = e^{-0.5 \cdot 2}
$$

### Step 2: Calculate $$ P(T > 1) $$
Using the formula:
$$
P(T > 1) = e^{-0.5 \cdot 1}
$$

Now, I will compute these values.
Calculate the value by following steps:
- step0: Calculate:
  $$e^{-0.5\times 2}$$
- step1: Multiply the numbers:
  $$e^{-1}$$
- step2: Express with a positive exponent:
  $$\frac{1}{e}$$
Calculate or simplify the expression $$ e^{-0.5 * 1} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$e^{-0.5\times 1}$$
- step1: Multiply:
  $$e^{-0.5}$$
- step2: Convert the expressions:
  $$e^{-\frac{1}{2}}$$
- step3: Express with a positive exponent:
  $$\frac{1}{e^{\frac{1}{2}}}$$
- step4: Simplify:
  $$\frac{1}{\sqrt{e}}$$
- step5: Multiply by the Conjugate:
  $$\frac{\sqrt{e}}{\sqrt{e}\times \sqrt{e}}$$
- step6: Multiply the numbers:
  $$\frac{\sqrt{e}}{e}$$
### Step 1: Probability that a repair time exceeds 2 hours
We calculated:
$$
P(T > 2) = e^{-0.5 \cdot 2} \approx 0.367879
$$

### Step 2: Probability that the repair time will take at least 4 hours given that the repair has been working on the machine for 3 hours
We calculated:
$$
P(T > 1) = e^{-0.5 \cdot 1} \approx 0.606531
$$

### Summary of Results
1. The probability that a repair time exceeds 2 hours is approximately $$ 0.367879 $$.
2. The probability that the repair time will take at least 4 hours given that the repair has been working on the machine for 3 hours is approximately $$ 0.606531 $$.
The probability that a repair time exceeds 2 hours is approximately 0.368. The probability that the repair time will take at least 4 hours given that it has already been working for 3 hours is approximately 0.607.

The time required to repair a machine is an exponential random variable with rate
owns/hour.
What is the probability that a repair time exceeds 2 hours?
What is the probability that the repait time will take at least 4 hours given that the repair
an been working on the machine for 3 hours?

Upstudy AI Solution

Answer

Solution

Step 1: Calculate

Step 2: Calculate

Step 1: Probability that a repair time exceeds 2 hours

Step 2: Probability that the repair time will take at least 4 hours given that the repair has been working on the machine for 3 hours

Summary of Results

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The time required to repair a machine is an exponential random variable with rate owns/hour. What is the probability that a repair time exceeds 2 hours? What is the probability that the repait time will take at least 4 hours given that the repair an been working on the machine for 3 hours?