Probability Distribution: Mean and Standard Deviation

Calculating the Mean of a Probability Distribution

• Formula: \mu = \Sigma(x \cdot P(x))

• Steps:

  1. Create a column for x \cdot P(x).

  2. Multiply each x by its corresponding P(x) and record the product.

  3. Sum all the x \cdot P(x) products; this sum is the mean, \mu.

Calculating the Standard Deviation of a Probability Distribution

• Formula: \sigma = \sqrt{\Sigma(x^2 \cdot P(x)) - \mu^2}

• Steps:

  1. Recall the unrounded mean (\mu).

  2. Add a column for x^2.

  3. Add a column for x^2 \cdot P(x).

  4. Square each x value and record it in the x^2 column.

  5. Multiply each x^2 value by its corresponding P(x) and record it in the x^2 \cdot P(x) column.

  6. Sum all the values in the x^2 \cdot P(x) column to get \Sigma(x^2 \cdot P(x)).

  7. Substitute values into the formula: \sigma = \sqrt{(\Sigma(x^2 \cdot P(x))) - \mu^2}.