Question 1163784: Consider a random variable with exponential distribution -1. Compute the probability for P (X>3).
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website! ```python?code_reference&code_event_index=2
import math
# Assuming lambda = 1
lambda_val = 1
x = 3
prob = math.exp(-lambda_val * x)
print(f"{prob=}")
```
```text?code_stdout&code_event_index=2
prob=0.049787068367863944
```
To compute the probability $P(X > 3)$ for a random variable $X$ with an exponential distribution, we first identify the parameter.
### 1. Identify the Parameter ($\lambda$)
In an exponential distribution, the parameter $\lambda$ (the rate) must be positive. The "-1" in your query likely refers to a rate of $\lambda = 1$ (where the dash is a separator).
The probability density function (PDF) is:
$$f(x; \lambda) = \lambda e^{-\lambda x} \text{ for } x \ge 0$$
### 2. Formula for $P(X > x)$
The probability that $X$ is greater than a specific value $x$ is given by the survival function:
$$P(X > x) = \int_{x}^{\infty} \lambda e^{-\lambda t} dt = e^{-\lambda x}$$
### 3. Calculation
Substituting $\lambda = 1$ and $x = 3$:
$$P(X > 3) = e^{-(1)(3)}$$
$$P(X > 3) = e^{-3}$$
Using the value of $e \approx 2.71828$:
$$P(X > 3) \approx 0.049787$$
**Final Answer:**
The probability is **$e^{-3}$**, which is approximately **$0.0498$** (or **4.98%**).
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