Complete Question
The complete question is shown on the first uploaded image
Answer:
The standard deviation is [tex]\sigma = 2.45[/tex]
Step-by-step explanation:
From the given data we can compute the expected mean for each random values as follows
[tex]E(X) = \sum [ X * P(X = x )]\\\\ X \ \ \ \ \ \ X* P(X =x )\\ 0 \ \ \ \ \ \ \ \ \ \ 0* 0.4 = 0 \\ 2 \ \ \ \ \ \ \ \ \ \ 2 * 0.3 = 0.6 \\ 4 \ \ \ \ \ \ \ \ \ \ 4 * 0.2 = 0.8\\ 6 \ \ \ \ \ \ \ \ \ \ 6* 0.1 = 0.6[/tex]
So
[tex]E(x) = 0 + 0.6 + 0.8 + 0.6[/tex]
[tex]E(x) = 2[/tex]
The
[tex]E(X^2) = \sum [ X^2 * P(X = x )]\\\\ X \ \ \ \ \ \ \ \ \ \ X^2 * P(X=x ) \\ 0 \ \ \ \ \ \ \ \ \ \ 0^2 * 0.4 = 0 \\ 2 \ \ \ \ \ \ \ \ \ \ 2^2 * 0.3 = 12 \\ 4 \ \ \ \ \ \ \ \ \ \ 4^2 * 0.2 = 3.2 \\ 6 \ \ \ \ \ \ \ \ \ \ 6^2 * 0.1 = 3.6[/tex]
So
[tex]E(X^2) = 0 + 1.2 + 3.2 + 3.6[/tex]
[tex]E(X^2) = 8[/tex]
Now the variance is mathematically evaluated as
[tex]Var (X) = E(X^2 ) -[E(X]^2[/tex]
Substituting value
[tex]Var (X) = 8-4[/tex]
[tex]Var (X) = 6[/tex]
The standard deviation is mathematically evaluated as
[tex]\sigma = \sqrt{Var(x)}[/tex]
[tex]\sigma = \sqrt{4}[/tex]
[tex]\sigma = 2[/tex]
