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How do you find the probability of a specific value when you only have the expected value and the function's variance?

For example, I'm asked to find

a) $P\{X = 4\}$ and

b) $P\{X>12\}$

If $E[X] = 7$ and $Var(X) = 2.1$

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    You can't, unless you know the distribution of $X$; at best you can provide a bound for the probability. Are you given that $X$ is normally distributed?2012-10-14
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    Or perhaps that it is uniformly distributed, or something like that?2012-10-14
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    No, only that $X$ is a binomial random variable.2012-10-14

1 Answers 1

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You know that $X \sim Binomial(n,p)$. Hence

$$ P(X=4)=\binom{n}{4}p^{4}(1-p)^{n-4}\\ P(X \geq 13)=\sum_{k=13}^{n}\binom{n}{k}p^{k}(1-p)^{n-k} $$ The latte by the way does not exist in closed form, so you will need to approximate it, e.g. with Markov or Chernoff bounds.

OK, the most interesting part is how to find these paramters, $p$ and $n$. If the variable is Binomial, we know that $$ \mathbf{E}X=np\\ \mathbf{Var}X=np(1-p) $$ You have these tow values above, hence you need to solve the system of equations: $$ np=7\\ np(1-p)=2.1 $$ After you obtain the values for $n,p$ plug them in the equations above.

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    I got n = 10, p = .7 Everything adds up. Thanks for the awesome explanation2012-10-15
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    wait- I got stuck. In the second binomial, k is bigger than n, I'm getting a math error when I do 10 choose 13. ??2012-10-15
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    you still online? Can you help me?2012-10-15
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    If $n=10$ then clearly $P(X>12)=0$2012-10-15
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    The way I understand this problem is that there is perhaps a dice with who knows how many sides that is rolled. All we now is that the Expected value is 7. I thought n referred to the number of times its thrown. There still should be a chance to get the 12 or higher. You understand what I'm saying. Where I am I missing something?2012-10-15
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    $n$ is the number of experiments, $X$ is the number of times of getting some 'success', e.g. 6 or '3 or more' or whatever. Obviously in 10 trials it is impossible to get $X$ 12 times, so P(X>12) is certainly 0. Check the values for EX and VarX2012-10-15