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To understand how to calculate the mean of a probability distribution, you need to follow a straightforward formula. The mean, often denoted as \( \text{E}(x) \), represents the expected value of the random variable. In simpler terms, it's like finding the center or average value.

The formula to find the mean for a discrete random variable is given by:

\( \text{Mean} = \text{E}(x) = \sum (x \times P(x)) \).

Let's break this down using our example. Here are the given values:

• \( x = 0, 1, 2, 3, 4, 5 \)
• \( P(x) = 0.031, 0.156, 0.313, 0.313, 0.156, 0.031 \)

To calculate the mean, we multiply each possible value of \( x \) by its corresponding probability \( P(x) \) and then add them all together:

\( \text{Mean} = 0 \times 0.031 + 1 \times 0.156 + 2 \times 0.313 + 3 \times 0.313 + 4 \times 0.156 + 5 \times 0.031 = 2.5 \).

This result means that on average, out of 5 children, 2.5 are expected to inherit the X-linked genetic disorder.

The standard deviation measures the spread or dispersion of a set of values. It tells us how much the values differ from the mean.

To find the standard deviation of a discrete random variable, we first need to calculate the variance. The variance, denoted as \( \text{Var}(x) \), is the average of the squared differences from the mean.
Here's how you calculate it step-by-step:

1. Calculate \( \text{E}(x^2) \): This is the expected value of \( x^2 \).
Formula: \( \text{E}(x^2) = \sum (x^2 \times P(x)) \).

2. Use the variance formula: \( \text{Variance} = \text{E}(x^2) - ( \text{E}(x) )^2 \).

3. Calculate the standard deviation: It's the square root of the variance.
Formula: \( \text{Standard Deviation} = \sqrt{\text{Variance}} \).

Let's apply these steps to our example:

• \( \text{E}(x^2) = 0^2 \times 0.031 + 1^2 \times 0.156 + 2^2 \times 0.313 + 3^2 \times 0.313 + 4^2 \times 0.156 + 5^2 \times 0.031 \)

\( = 0 + 0.156 + 1.252 + 2.817 + 2.496 + 0.775 = 7.496 \)

Next, use the variance formula:

\( \text{Variance} = 7.496 - (2.5^2) = 1.246 \).

Finally, the standard deviation:

\( \text{Standard Deviation} = \sqrt{1.246} = 1.116 \).

So, the standard deviation of our variable is approximately 1.116, indicating how much the number of children who inherit the disorder varies from the mean.

A discrete random variable is a type of variable that can take only specific and distinct values. Unlike continuous variables, which can take any value within a range, discrete variables are countable.

In our example, the random variable \( x \) represents the number of children who inherit the X-linked genetic disorder. Here are the possible values it can take:

• \( x = 0, 1, 2, 3, 4, 5 \)

To check if a given set of values forms a probability distribution, we need to meet two main criteria:

1. The sum of all probabilities \( P(x) \) should be equal to 1.
2. Each individual probability value should be between 0 and 1, inclusive.

Let's verify these criteria with our provided values:

• Sum of probabilities: \( 0.031 + 0.156 + 0.313 + 0.313 + 0.156 + 0.031 = 1.000 \).

This satisfies the first requirement.

Next, check the individual probabilities:

• \( 0.031, 0.156, 0.313, 0.313, 0.156, 0.031 \) are all between 0 and 1.

Both conditions are satisfied. Therefore, this is a valid probability distribution.

Understanding discrete random variables is essential. They help us model real-world situations where outcomes are distinct and countable, like in our genetic disorder example.