Normal Distribution and Sum of Random Variables (Summing Normal Random Variables)

Normal Distribution Properties

• The normal distribution is unique because the sum of normal random variables is also a normal random variable.

• This property is useful when analyzing samples of data and combining samples.

Real-World Example: Statistics Textbook Production and Delivery

• Consider the production and delivery of statistics textbooks.

• Production time is normally distributed with a mean of 7 weeks and a standard deviation of 2 weeks.

• Delivery time is also normally distributed with a mean of 1 week and a standard deviation of 0.5 weeks.

Combining Normal Distributions

• Given two normally distributed random variables, $x$ and $y$, with means $\mu<em>1$ and $\mu</em>2$, and variances $\sigma<em>1^2$ and $\sigma</em>2^2$:

  • The mean of the sum of the random variables is the sum of the means: $\mu = \mu<em>1 + \mu</em>2$.

  • The variance of the sum of the random variables is the sum of the variances: $\sigma^2 = \sigma<em>1^2 + \sigma</em>2^2$.

  • It is important to note that standard deviations cannot be directly added.

Applying to the Textbook Example

• Delivery time:

  • Mean: 1 week

  • Standard deviation: 0.5 weeks

  • Variance: $(0.5)^2 = 0.25$ weeks

• Combined production and delivery:

  • Total mean: $7 + 1 = 8$ weeks

  • Total variance: $4 + 0.25 = 4.25$ weeks

  • Total standard deviation: $\sqrt{4.25} \approx 2.06$ weeks

Calculating Probabilities

• We can use this combined distribution to calculate probabilities.

• For example, the probability that production and delivery take place within ten weeks.

Using Excel

• Use the NORM.DIST function in Excel.

• $x = 10$ weeks

• Mean = 8 weeks

• Standard deviation = 2.06 weeks

• Cumulative = TRUE

• The probability that the textbook is both produced and delivered within ten weeks is approximately 0.834.

Conclusion

• Adding two normal distributions results in another normal distribution.

• This property allows for the calculation of probabilities using normal distribution functions.