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Probability distribution: consists of all the possible values for a random variable and corresponding probability of the value. The probabilities are determined ...
Typology: Study notes
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Analyze: X could be 0, 1, 2 , or 3. If we put all possible values and there corresponding probabilities in a table, we will have the below:
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Number of boys X
Probability P(x)
Tables like this are called probability distribution. Probability distribution : consists of all the possible values for a random variable and corresponding probability of the value. The probabilities are determined by calssical method or empirical method. Probability histogram: is a histogram in which the horizontal axis indicates the values of a random variable and vertical axis indicates the corresponding probability.
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0 1 2 3 More
probability P(x)
X
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Variance for a discrete random variable:
these two formulas can be used either one.
standard deviation of a discrete random:
Ex: The below table shows a probability distribution for the
random variable X which represents the number of shots made for a basketball player to shoot three free throws, X could be 0,1,2, or 3.
2 2 2 2
(^) x x
P(x) 0.01 0.10 0.38 0.
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Q:calculate the mean, variance and standard deviation for
the discrete variable.
Ans: Mean
=00.01+10.1+20.38+30.51=2.
x P(x) 0 0.01 (0-2.39)^2 *0.01=0. 1 0.1 (1-2.39)^2 *0.1=0. 2 0.38 (2-2.39)^2 *0.38=0. 3 0.51 (3-2.39)^2 *0.51=0.
( x X )^2 * p ( x ) 0. 4979
( ) * ( ) 0. 4979 0. 5
X x X P x
(^) X ^2 0. 4979 0. 7
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Binominal probability distribution function (PDF):
The probability of obtaining x successes in n independent
trials of a binomial experiment is given by:
where p is the probability of success.
Mean(expected value) and standard deviation of a
binomial distribution:
A binominal experiment with n independent trials and
probability of success P has a mean and standard deviation given by the formulas:
P x C p p x n x n x ( ) (^) n x ( 1 ) 0 , 1 , 2 ,...,
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Example 3, 4 shows a good application on binominal
probability
Example 6 shows how to use binominal table to find the
probability
Example 7 shows how to create a histogram for a
binominal distribution.
Let’s take more time to learn these four examples.