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So if you want to include P (X=k) where k is the lower limit of some interval, you'll go ½ an interval further, down to k -½, but if k is the upper limit of your interval for the pmf, you'll go up to k +½ on the approximation.
The Central Limit Theorem in Statistics states that as the sample size increases and its variance is finite, then the distribution of the sample mean approaches the normal distribution, irrespective of the shape of the population distribution.
The central limit theorem tells us that even 62 is sufficient; it gives a lower value because we know something about the distribution we are bounding, namely that it is a mean of IID random variables.
It is important for you to understand when to use the central limit theorem. If you are being asked to find the probability of the mean, use the clt for the mean. If you are being asked to find the probability of a sum or total, use the clt for sums. This also applies to percentiles for means and sums.
We add 0.5 if we are looking for the probability that is less than or equal to that number. We subtract 0.5 if we are looking for the probability that is greater than or equal to that number.
There are two ways that you could state the central limit theorem. Either that the sum of IID random variables is normally distributed, or that the average of IID random variables is normally distributed.
To correct for discreteness of x. Deviations can be seen especially at tails. Figure 6-21 Normal probability plots indicating a non-normal distribution. The X are independent random variables. has the same probability distribution. A statistic is any function of the observations in a random sample.
It is important for you to understand when to use the central limit theorem. If you are being asked to find the probability of the mean, use the clt for the means.
In order to get the best approximation, add 0.5 to x or subtract 0.5 from x (use x + 0.5 or x – 0.5). The number 0.5 is called the continuity correction factor and is used in the following example.
In summary, the Central Limit Theorem explains that both the sample mean of IID variables is normal (regardless of what distribution the IID variables came from) and that the sum of equally weighted IID random variables is normal (again, regardless of the underlying distribution).