4. Probability Distributions: Common Distributions
Expected Value and Variance of a Binomial Random Variable
Expected Value of a Binomial Random Variable
To calculate the expected value of a binomial random variable #X\sim B(n,p)#, use the following formula:
\[\mathbb{E}[X] = n\cdot p\]
Suppose #X# has a binomial distribution with parameters #n=10# and #p=0.4#.
Calculate the expected value of #X#.
Calculate the expected value of #X#.
#\mathbb{E}[X]=4.0#
To calculate the expected value of a binomial random variable #X\sim B(n,p)#, use the following formula:
\[\mathbb{E}[X]=n \cdot p=10 \cdot 0.4 = 4.0\]
To calculate the expected value of a binomial random variable #X\sim B(n,p)#, use the following formula:
\[\mathbb{E}[X]=n \cdot p=10 \cdot 0.4 = 4.0\]
#\phantom{0}#
Variance and Standard Deviation of a Binomial Random Variable
To calculate the variance of a binomial random variable #X\sim B(n,p)#, use the following formula:
\[\sigma^2 = n\cdot p \cdot (1-p)\]
To calculate the standard deviation, take the positive square root of the variance:
\[\sigma = \sqrt{\sigma^2} = \sqrt{n\cdot p \cdot (1-p)}\]
#\sigma^2=2.75#
To calculate the variance of a binomial random variable #X\sim B(n,p)#, use the following formula:
\[\sigma^2 = n\cdot p \cdot (1-p) = 11 \cdot 0.5 \cdot (1-0.5) = 2.75\]
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