Central limit theorem for sums

Definition

The Central Limit Theorem for Sums states that the distribution of the sum of a large number of independent, identically distributed random variables approaches a normal distribution, regardless of the original distribution. This theorem allows for approximations of sums using the normal distribution when sample sizes are sufficiently large.

5 Must Know Facts For Your Next Test

1. The mean of the sum is equal to the sum of the means of individual random variables.
2. The variance of the sum is equal to the sum of the variances of individual random variables.
3. For large sample sizes (typically n > 30), the sum can be approximated by a normal distribution.
4. Sums involving independent and identically distributed (i.i.d.) random variables are central to this theorem.
5. The theorem is crucial in practical applications like confidence intervals and hypothesis testing.

Review Questions

• What conditions must be met for the Central Limit Theorem for Sums to apply?
• How does one approximate the mean and variance in the context of this theorem?
• Why is it important that random variables are independent and identically distributed?

Related terms

Normal Distribution: A continuous probability distribution characterized by a symmetric, bell-shaped curve.

Variance: A measure of how much values in a data set differ from the mean.

Independent Random Variables: Random variables whose outcomes do not affect each other.