To find the expected values for two types of random variables—discrete and continuous—we need to use different methods because they are quite different from each other.
What They Are: A discrete random variable can only take specific, countable values. Think of it like rolling a dice. You can only get a 1, 2, 3, 4, 5, or 6.
How to Find Expected Value:
We use this formula:
[
E(X) = \sum_{i} x_i P(X = x_i)
]
Here, (x_i) represents the possible values, and (P(X = x_i)) is the chance of each value happening.
Let's See an Example:
If you roll a dice, the expected value can be calculated like this:
[
E(X) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6) = 3.5
]
This means that over many rolls, you would expect the average result to be about 3.5.
What They Are: A continuous random variable can take on endless values within a range. Imagine measuring height—people can be any height between, say, 4 feet and 7 feet.
How to Find Expected Value:
For this, we use a different formula:
[
E(X) = \int_{-\infty}^{\infty} x f(x) , dx
]
In this formula, (f(x)) is called the probability density function, which helps us understand how likely different values are.
Let's Look at an Example:
If we have a uniform distribution between two numbers (a) and (b) (like all heights between 5 and 6 feet), the expected value is found with this formula:
[
E(X) = \frac{a + b}{2}
]
So, if (a) is 5 and (b) is 6, then the expected value would be:
[
E(X) = \frac{5 + 6}{2} = 5.5
]
This tells us that, on average, someone would be around 5.5 feet tall in that range.
In summary, while discrete variables deal with specific numbers we can count, continuous variables deal with a whole range of possible values. Both types have their own way of calculating expected values!
To find the expected values for two types of random variables—discrete and continuous—we need to use different methods because they are quite different from each other.
What They Are: A discrete random variable can only take specific, countable values. Think of it like rolling a dice. You can only get a 1, 2, 3, 4, 5, or 6.
How to Find Expected Value:
We use this formula:
[
E(X) = \sum_{i} x_i P(X = x_i)
]
Here, (x_i) represents the possible values, and (P(X = x_i)) is the chance of each value happening.
Let's See an Example:
If you roll a dice, the expected value can be calculated like this:
[
E(X) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6) = 3.5
]
This means that over many rolls, you would expect the average result to be about 3.5.
What They Are: A continuous random variable can take on endless values within a range. Imagine measuring height—people can be any height between, say, 4 feet and 7 feet.
How to Find Expected Value:
For this, we use a different formula:
[
E(X) = \int_{-\infty}^{\infty} x f(x) , dx
]
In this formula, (f(x)) is called the probability density function, which helps us understand how likely different values are.
Let's Look at an Example:
If we have a uniform distribution between two numbers (a) and (b) (like all heights between 5 and 6 feet), the expected value is found with this formula:
[
E(X) = \frac{a + b}{2}
]
So, if (a) is 5 and (b) is 6, then the expected value would be:
[
E(X) = \frac{5 + 6}{2} = 5.5
]
This tells us that, on average, someone would be around 5.5 feet tall in that range.
In summary, while discrete variables deal with specific numbers we can count, continuous variables deal with a whole range of possible values. Both types have their own way of calculating expected values!