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The binomial distribution is a pdf for discrete random variables. The binomial distribution is used in trials with only two possible outcomes: success or failure. The probability of success is represented by p. The probability of failure is represented by 1-p or q where q = 1-p. The binomial distribution is as follows:
| P( X = k ) | = |
|
( 1 - p )(n-k)pk |
where:
| n | = | number of trials | ||||
| k | = | number of successes | ||||
| n-k | = | number of failures | ||||
| p | = | probability of success | ||||
| 1-p | = | probability of failure | ||||
|
= | the number of possible successful outcomes out of n trials |
Examples:
If a single dice is rolled 3 times, what are the chances of getting a 2 in one roll.
| P( k = 1 ) | = |
|
( 1 - 1/6)(3-1)(1/6)1 | ||||
| = | 3( 5/6)2(1/6) | ||||||
| = | .347 | ||||||
If a coin is tossed 7 times, what is the probability of 3 heads occuring.
| P( k = 3) | = |
|
( 1 - 1/2)(7-3)(1/2)3 | ||||
| = |
|
( 1/2 )4(1/2)3 | |||||
| = |
|
( 1/2 )7 | |||||
| = | .273 |
To find the probability of a random variable X being less than or equal to a certain value, a special function called the cumulative distribution function is used. It is abbreviated as cdf. In the case of a discrete random variable, all the probabilities are added together for all the values less than the value of the random variable.
The cumulative distribution function for any random variable is written as:
F(a) =P( X ≤ a )
where F(a) is the cdf.
For a discrete random variable X, the cdf is represented as:
F(a) = ∑xi < a p( xi )
where xi is each value less than a and a is the value of the random variable.
In the case where a random variable is continuous, the cdf has this form:
F(y) = ∫-∞→y f(t)dt
where y is the value of random variable y and f(t) is the probability density function for y.
cdfs are monotonically non-decreasing which means that the value of the cdf for every value of the random variable is greater than for the lesser values of the random variable:
F(a) ≤ F(b) when a < b
also,
0 ≤ F(a) ≤ 1 when X = a
If y is a continuous random variable that has a pdf of fY(y) and a cdf of FY(y) then:
fY(y) = F'Y(y)
In other words, the pdf is the derivative of the cdf.
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