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MSV 3: Most Likely Value

www.making-statistics-vital.co.uk. MSV 3: Most Likely Value. Dec is about to take n penalties. The probability he is successful with each penalty is p , where p is a constant, independently of the other attempts, and he knows what n and p are.

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MSV 3: Most Likely Value

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  1. www.making-statistics-vital.co.uk MSV 3: Most Likely Value

  2. Dec is about to take n penalties. The probability he is successful with each penalty is p, where p is a constant, independently of the other attempts, and he knows what n and p are.

  3. Dec calls ‘the number of successful penalties he is about to achieve’ X, and he wants to know what the most likely value for X is. His friend Ant offers him some statistical advice.

  4. ‘Dec, Rule One is this: your most likely value for X must be a whole number, but the expectation of Xneed not be a whole number.’ ‘But Ant, surely the number you ‘expect’ will be a whole number? ‘Suppose,’ says Ant, ‘you have a biased six-sided dice, where the probability of getting a 6 is five times the probability of getting each of the other numbers.’

  5. ‘It’s obvious that the most likely value we get when we roll a dice here is 6. What is the expectation here?’ ‘Well, 10a = 1, so a = 0.1, and E(X) = 45a = 4.5.’ ‘So the most likely value is a whole number, but the expectation is not.’ ‘But you can’t actually roll 4.5!’ says Dec. ‘The expectation is ‘an average value’,’ says Ant, ‘so you don’t need to be able to!’

  6. ‘You’ve convinced me,’ says Dec. So what is Rule Two?’ ‘Ah, now Rule Two,’ Ant says, ‘tells you that if the expectation of XIS a whole number, then that will be the most likely value.’

  7. ‘That seems reasonable,’ says Dec. ‘What happens if the expectation of Xis not a whole number?’ ‘Then you need Rule Three,’ says Ant. ‘The most likely value of X is one of the whole numbers on either side of the expectation of X.’

  8. ‘Just work out the probabilities for the whole numbers on either side of the expectation of X, and pick the one that gives you the larger value. This will be the most likely value.

  9. Good news: Ant is offering reliable advice! How do we know this? Let’s now evaluate Rules One, Two, andThree.

  10. Answers Maybe the best thing to do to start with is to play around with an applet that shows the Binomial probabilities for various n and p. A helpful address is the following page at the excellent Waldomaths site at: http://www.waldomaths.com/PoissBin1NL.jsp

  11. Rule One is definitely true. After a little experimentation, it seems that Rule Two is likely to be true. What about Rule Three? It would seem likely that we find the most likely value by rounding the expectation of X to the nearest whole number. This, however, fails to stand up. It is not too hard to find a counter-example.

  12. For example, X ~ B(20,32/75) gives E(X) = 8.5333. So roundingwould suggest that the most likely value for X is 9. But P(X = 8) = 0.17452… and P(X = 9) = 0.17317… So the most likely value for X here is actually 8.

  13. If np is a whole number, then as q < 1, then by *, np is the most likely value, so Rule Two is true. The inequality * also tells us that Rule Three is true.

  14. www.making-statistics-vital.co.uk is written by Jonny Griffiths hello@jonny-griffiths.net

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