Question

5.1 basics of probability distributions. compute the mean and standard deviation of a variable.

x	p(x)
1	0.05
4	0.25
7	0.15
12	0.55

find the mean and standard deviation of this probability distribution.
give your answer to at least 2 decimal places.
mean (μ):
standard deviation (σ):
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Explanation:

Step1: Calculate the mean formula

The mean $\mu$ of a discrete - probability distribution is given by $\mu=\sum_{i}x_{i}P(x_{i})$.
$\mu=(1\times0.05)+(4\times0.25)+(7\times0.15)+(12\times0.55)$

Step2: Compute the mean value

$\mu = 0.05+1 + 1.05+6.6$
$\mu=8.70$

Step3: Calculate the variance formula

The variance $\sigma^{2}=\sum_{i}(x_{i}-\mu)^{2}P(x_{i})$.
$(1 - 8.7)^{2}\times0.05+(4 - 8.7)^{2}\times0.25+(7 - 8.7)^{2}\times0.15+(12 - 8.7)^{2}\times0.55$
$=(-7.7)^{2}\times0.05+(-4.7)^{2}\times0.25+(-1.7)^{2}\times0.15+(3.3)^{2}\times0.55$
$=59.29\times0.05 + 22.09\times0.25+2.89\times0.15 + 10.89\times0.55$
$=2.9645+5.5225 + 0.4335+5.9895$
$=14.91$

Step4: Calculate the standard - deviation

The standard deviation $\sigma=\sqrt{\sigma^{2}}$.
$\sigma=\sqrt{14.91}\approx3.86$

Answer:

Mean ($\mu$): $8.70$
Standard Deviation ($\sigma$): $3.86$