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any one to help me with descriptions or to tell me where i can retrieve the answers for this kind of question. Describe in detail the circumstances in which the binomial distribution can be applied,and when can it be approximated by the normal or Gaussian probability density function?

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I think you can probably retrieve the answer to this question in your lecture notes. Also here and here. –  Elvis Dec 22 '11 at 10:42
    
If this is homework (as the phrasing of the question indicates very much), please add the tag (homework). –  Did Dec 22 '11 at 10:57
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4 Answers

Let me give you an example of where the binomial distribution applies:

Consider a supermarket. They sell eggs in packets of six. Their transport system from the farm to the shop is not very good, so some of the eggs get damaged on the way. If an egg gets damaged with probability $p$ then the number of damaged eggs in a packet is binomially distributed.

As for approximating it with a normal distribution, see here.

Hope this helps.

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Probably not. The binomial requires that the eggs break independently. Given the eggs are in packets if one breaks it is more likely that a neighbor will also break than the binomial proportion p for an individual egg. In your example there is positive correlation between neighboring eggs. –  Michael Chernick Aug 31 '12 at 15:42
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It can be applied whenever you're talking about the number of successes in a fixed number of independent trials, with the same fixed probability of success on each trial. (And "fixed" means not random.)

If $X$ has a binomial distribution with parameters $n$ and $p$ ($n = {}$number of trials and $p={}$probability of success on each trial) then $$ \Pr\left( X \le x \right) \approx \Pr\left( Y\le x+\frac12 \right) $$ where $Y$ is normally distributed with expected value $np$ (the same as the expected value of $X$ and standard deviation $\sqrt{np(1-p)}$ (the same as the standard deviation of $X$).

Notice that the event $(X\le x)$ is the same as the event $(X<x)$, and with $Y$ we just use the number half-way between those two. The approximation error is never more than about $0.03$ even when $n=1$, but there's a crude rule of thumb that to get good approximations, you should have $np\ge5$ and $n(1-p)\ge 5$.

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Here's a more concrete answer than my previous one. It's just one example.

Suppose $p\cdot 100\%$ of the voters will vote "yes" in next week's referendum. You take a random sample of 200 voters and ask whether they will vote "yes". The number who will vote "yes" is binomially distributed with parameters $200$ and $p$.

Often results of polls are reported with a "margin of error". If $60$ of the $200$ in the sample will vote "yes", then one can use $60/200=0.3$ as an estimate of $p$. And one can take $\sqrt{200(0.3)(1-0.3)}$ to be an estimate of the standard deviation. A normally distributed random variable is more than $1.96$ standard deviations away from its expected value only $5\%$ of the time, so one can use $0.3 \pm 1.96\cdot\sqrt{200(0.3)(1-0.3)}/200$ as a $95\%$ confidence interval for $p$. (This comes to $(0.23649,0.36351)$).)

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binomial distribution can be used in genetics to determine the probability the k out of n individuals will have a particular genotype in this case having that particular genotype is considered success binomial distribution also can be used in the quality control

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