Question

Find u if p = [x•P(x)]. Then, find o if o2 = {[x• P(x)] –H2 X P(x) 0 0.0005 1 0.0091 2 0.0648 3 0.2297 4 0.4072 5 0.2887 (Simplify your answer. Round to four decimal places as needed.) O= (Simplify your answer. Round to four decimal places as needed.)

Answer 1

X	P(X)
0	0.0005
1	0.0091
2	0.0648
3	0.2297
4	0.4072
5	0.2887

Now, we need to multiply the corresponding X outcomes with the corresponding probabilities, in order to compute the population mean μ:

X	p(X)	X⋅p(X)
0	0.0005	0⋅0.0005=0
1	0.0091	1⋅0.0091=0.0091
2	0.0648	2⋅0.0648=0.1296
3	0.2297	3⋅0.2297=0.6891
4	0.4072	4⋅0.4072=1.6288
5	0.2887	5⋅0.2887=1.4435

Therefore, the population mean is calculated as follows:

$\begin{array}{ccl} \mu & = & \sum_{i=1}^{n} X_i p(X_i) \\ \\ & = & \displaystyle 0\cdot 0.0005+1\cdot 0.0091+2\cdot 0.0648+3\cdot 0.2297+4\cdot 0.4072+5\cdot 0.2887 \\ \\ & = & \displaystyle 3.9001 \end{array}$

Now, we need to multiply the corresponding squares $X^2$ of the outcomes with the corresponding probabilities, in order to compute the population variance $\sigma^2$ :

X	p(X)	X⋅p(X)	X^2⋅p(X)
0	0.0005	0⋅0.0005=0	$0^2 \cdot 0.0005 = 0$
1	0.0091	1⋅0.0091=0.0091	12. 0.0091 = 0.0091
2	0.0648	2⋅0.0648=0.1296	$2^2 \cdot 0.0648 = 0.2592$
3	0.2297	3⋅0.2297=0.6891	$3^2 \cdot 0.2297 = 2.0673$
4	0.4072	4⋅0.4072=1.6288	$4^2 \cdot 0.4072 = 6.5152$
5	0.2887	5⋅0.2887=1.4435	$5^2 \cdot 0.2887 = 7.2175$

Therefore, we first compute the expected value of $X^2$ :

$\begin{array}{ccl} E(X^2) & = & \sum_{i=1}^{n} X_i^2 p(X_i) \\ \\ & = & \displaystyle 0^2 \cdot 0.0005+1^2 \cdot 0.0091+2^2 \cdot 0.0648+3^2 \cdot 0.2297+4^2 \cdot 0.4072+5^2 \cdot 0.2887 \\ \\ & = & \displaystyle 16.0683 \end{array}$

Therefore, the population variance is computed as follows:

$\begin{array}{ccl} \sigma^2 &= & E(X^2) - E(X)^2 \\ \\ & = & \displaystyle 16.0683 - 3.9001^2 \\ \\ & = & \displaystyle 0.8575 \end{array}$

And finally, taking square root to the variance we get that the population standard deviation is $\sigma = \sqrt{ \sigma^2} = \sqrt{0.8575} = 0.926$

Therefore, based on the discrete distribution, the population mean is μ=3.9001 and the population standard deviation is σ=0.926