Knowing that each die has six possible values (1, 2, 3, 4, 5, 6), when we record the average of the 10 dice over time, we would expect the number to start approximating 3.5 (the average of all possible values). Upon doing so, we will discover that the probability distribution of the sample average approximates the normal distribution even though a single roll of a fair die follows a discrete uniform distribution. We repeat rolling the die 50 times until we will have 50 data points. Each time we roll all 10 die together we record the average of the 10 die.
Let us assume we have 10 fair die at hand.
The following example will explain this further. According to the LLN, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. Consider the law of large numbers (LLN)-It is a theorem that describes the result of performing the same experiment a large number of times. It is one of the basic assumptions in Central Limit Theorem. Independent and Identically DistributedĪ sequence of random variables is independent and identically distributed (i.i.d.) if each random variable is independent of others and has the same probability distribution as others. When n increases, the sample average of the n random variables is approximately normally distributed, with the mean equal to μ and variance equal to σ2/n, regardless of the common distribution Yi follows where i = 1, 2. random variables, each of which has finite mean μ and variance σ2, where σ2 > 0. It states a condition under which the mean of a large number of independent and identically-distributed random variables, each of which has a finite mean and variance, would be approximately normally distributed. The Central Limit Theorem is one of the fundamental theorems of probability theory.
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