a phone buttery below as a random variable x with the The life of in hours Is Modeled probability density Runction: F(t) (t+2}3
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Understanding the Battery Life Model
The life of a phone battery in hours is modeled using a random variable \( X \). The probability density function (PDF) associated with this random variable is defined as follows:
\[
f(t) =
\begin{cases}
\frac{2}{(t+2)^3} & \text{for } t \geq 0 \\
0 & \text{for } t < 0
\end{cases}
\]
#### Explanation of the Function
- **For \( t \geq 0 \):** The PDF is defined as \( \frac{2}{(t+2)^3} \). This implies that the probability of the battery life being any specific value \( t \) depends on this expression. The function decreases as \( t \) increases, indicating that longer battery life durations are less probable.
- **For \( t < 0 \):** The PDF is zero. This makes intuitive sense because negative time doesn't have a physical interpretation in this context, so the probability of a negative battery life is zero.
This PDF helps in understanding the likelihood of various battery life durations and is useful for statistical analysis and prediction of battery performance over time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb4f8469-eda0-4bfd-b67e-0d596825d7c6%2Ffbcf0cf3-8745-4b44-ada2-c7fd629ce043%2Ff5b947_processed.jpeg&w=3840&q=75)
Given
Step by step
Solved in 2 steps
