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=13# and #p=0.8#.
Calculate the expected value of #X#.
Calculate the expected value of #X#.
#\mathbb{E}[X]=10.4#
To calculate the expected value of a binomial random variable #X\sim B(n,p)#, use the following formula:
\[\mathbb{E}[X]=n \cdot p=13 \cdot 0.8 = 10.4\]
To calculate the expected value of a binomial random variable #X\sim B(n,p)#, use the following formula:
\[\mathbb{E}[X]=n \cdot p=13 \cdot 0.8 = 10.4\]
#\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=2.08#
To calculate the variance of a binomial random variable #X\sim B(n,p)#, use the following formula:
\[\sigma^2 = n\cdot p \cdot (1-p) = 13 \cdot 0.8 \cdot (1-0.8) = 2.08\]
Unlock full access
