Question

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

x	p(x)
5	0.15
8	0.3
11	0.45

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 - random variable is given by $\mu=\sum_{i}x_{i}P(x_{i})$.
\[

$$\begin{align*} \mu&=(2\times0.1)+(5\times0.15)+(8\times0.3)+(11\times0.45)\\ &=0.2 + 0.75+2.4 + 4.95\\ &=8.3 \end{align*}$$

\]

Step2: Calculate the variance formula

The variance $\sigma^{2}=\sum_{i}(x_{i}-\mu)^{2}P(x_{i})$.
\[

$$\begin{align*} &(2 - 8.3)^{2}\times0.1+(5 - 8.3)^{2}\times0.15+(8 - 8.3)^{2}\times0.3+(11 - 8.3)^{2}\times0.45\\ &=(-6.3)^{2}\times0.1+(-3.3)^{2}\times0.15+(-0.3)^{2}\times0.3+(2.7)^{2}\times0.45\\ &=39.69\times0.1 + 10.89\times0.15+0.09\times0.3 + 7.29\times0.45\\ &=3.969+1.6335 + 0.027+3.2805\\ &=8.91 \end{align*}$$

\]

Step3: Calculate the standard - deviation formula

The standard deviation $\sigma=\sqrt{\sigma^{2}}$. Since $\sigma^{2}=8.91$, then $\sigma=\sqrt{8.91}\approx2.98$.

Answer:

Mean ($\mu$): $8.30$
Standard Deviation ($\sigma$): $2.98$