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This might be a dumb question.. but I was wondering if someone can help me out with the concept expectation. This question started from trying to understand bias vs consistent. So when we roll a dice, we usually say that the expected value is 3.5 I sort of assumed that getting 3.5 means that we roll the dice infinite number of times. The average value of rolling a dice 10 times will not be 3.5. The more times we roll, the closer we get to 3.5 so I thought 3.5 meant that we're rolling it infinite (or realistically close to infinite) number of times. But, when learning the difference between bias vs consistent estimators, there was this estimator that is biased but consistent. That to me is sort of confusing because going back to the dice example, the expected value of 3.5, in my thought, meant that we are already rolling the dice infinite number of times.. Can some one help me out with this please? Thank you!

chunguc1004
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    "Expected value" is defined after a "random variable" is defined, and it is a theoretical value that does not depend on the "rolling". – Zhanxiong Feb 13 '23 at 21:43
  • What you described is law of large numbers: that the sample average of a number of dies converges to 3.5, the theoretical expected value. – Zhanxiong Feb 13 '23 at 21:48
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    What @Zhanxiong wrote is correct. There is no need to have data or samples in order to calculate an expected value. Because the law(s) of large numbers are true, however, I do think the idea of rolling a die over and over gives an intuition about what the expected value of a die roll means. – Dave Feb 13 '23 at 21:51
  • The die can be modeled by placing the values $1,2,\ldots, 6$ on tickets in a box. Its roll is modeled by drawing one ticket out blindly. By definition, the expectation is the average of the tickets. Computing it only requires summing all six values and dividing by six. No infinities are needed. See, inter alia, https://stats.stackexchange.com/a/54894/919 for more discussion. – whuber Feb 13 '23 at 21:58
  • @Zhanxiong Thank you for your reply! So.. the reason why I'm confused is that the "theoretical value" cannot be reached if we don't assume having a large number of samples right?.. – chunguc1004 Feb 13 '23 at 23:16
  • @chunguc1004 The theoretical value is determined by probability theory (and the assumption that each face is landing up with equal chance). Usually the procedure of calculating "theoretical value" is taught in a probability course. Have you taken it before? – Zhanxiong Feb 13 '23 at 23:24
  • In the case of a sample average of $3.5$, it can be reached with two rolls, though other values are also possible. But it the expectation of one roll. – Henry Feb 14 '23 at 01:15

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