IGNOU MTE 11 2018 - X is a random variable taking values 0 and 1 with respective probabili

Question

X is a random variable taking values 0 and 1 with respective probabilities q and p. A sample X1 , X2 ,...., Xn of size n is taken from the distribution.If $\sum_{i=1}^{n}X_{i}$ , shows that the bias of the estimator tends to zero as n tends to infinity.

14 Feb 2018

Answer :

Word Count : 352

To solve this numerically, let's first set up the problem mathematically: Let $ X $ be a random variable taking values 0 and 1 with probabilities $ q $ and $ p $, respectively. So, the probability distribution of $ X $ is: \[ P(X = 1) = p \quad \text{and} \quad P(X = 0) = q = 1 - p \] Now, we have a sample of size $ n $, denoted by $ X_1, X_2, ..., X_n $, taken from this distribution. We want to consider the sum of these random variables: \[ S_n = \sum_{i=1}^{n} X_i \] This sum represents the total number of successes in $ n $ trials. Each $ X_i $ is an independent Bernoulli random variable with success probability $ p $. The expectation of the sum $ S_n $ is given by: \[

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