Suppose X takes the value 1 with probability P, and the value 0 with probability (1-P). The Probability distribution of X is P(X) 1-P 1 P The expected value of X, E(X) is:
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!
![### Probability Distribution and Expected Value Calculation
**Problem Statement:**
Suppose \( X \) takes the value 1 with probability \( P \), and the value 0 with probability \( (1-P) \).
#### Probability Distribution of \( X \):
| \( X \) | \( P(X) \) |
|--------|------------|
| 0 | \( 1-P \) |
| 1 | \( P \) |
The expected value of \( X \), \( E(X) \), is:
#### Multiple Choice Options:
- ○ \( P \)
- ○ \( nP \)
- ○ \( P(1-P) \)
- ○ \( \frac{1-P}{P} \)
**Explanation of Calculation:**
The expected value \( E(X) \) of a discrete random variable \( X \) is given by the sum of the product of each value that \( X \) can take and its corresponding probability. Mathematically,
\[
E(X) = \sum (x_i \cdot P(x_i))
\]
For this particular case:
\[
E(X) = 0 \cdot (1-P) + 1 \cdot P
\]
\[
E(X) = P
\]
Thus, the correct answer is:
- ○ \( P \)
This educational example demonstrates the fundamental concept of a probability distribution and how to calculate the expected value, which is a measure of the central tendency of a random variable.
#### Graphical Representation:
There isn't a graphical representation present in this text. The probability distribution provided is in a tabular form showing the probabilities for each corresponding value of \( X \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd529b6da-11b0-40e5-85fb-f211e59fb34e%2Fe5820067-c5bb-4c38-8d9c-cb7dcc220ae2%2F4cwepy.png&w=3840&q=75)
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