1.
X P(X)
0 0.2
1 0.3
2 0.2
3 0.2
4 0.1
μ = ∑ x ⋅ p ( x ) μ = 0 ⋅ 0.2 + 1 ⋅ 0.3 + 2 ⋅ 0.2 + 3 ⋅ 0.2 + 4 ⋅ 0.1 μ = 1.7 ∑ x 2 ⋅ p ( x ) = 0 2 ⋅ 0.2 + 1 2 ⋅ 0.3 + 2 2 ⋅ 0.2 + 3 2 ⋅ 0.2 + 4 2 ⋅ 0.1 ∑ x 2 ⋅ p ( x ) = 4.5 σ 2 = ∑ x 2 ⋅ p ( x ) − μ 2 = 4.5 − 1. 7 2 = 1.61 \begin{aligned}
&\mu=\sum x \cdot p(x) \\
&\mu=0 \cdot 0.2+1 \cdot 0.3+2 \cdot 0.2+3 \cdot 0.2+4 \cdot 0.1 \\
&\mu=1.7 \\
&\sum x^{2} \cdot p(x)=0^{2} \cdot 0.2+1^{2} \cdot 0.3+2^{2} \cdot 0.2+3^{2} \cdot 0.2+4^{2} \cdot 0.1 \\
&\sum x^{2} \cdot p(x)=4.5 \\
&\sigma^{2}=\sum x^{2} \cdot p(x)-\mu^{2}=4.5-1.7^{2}=1.61 \\
\end{aligned} μ = ∑ x ⋅ p ( x ) μ = 0 ⋅ 0.2 + 1 ⋅ 0.3 + 2 ⋅ 0.2 + 3 ⋅ 0.2 + 4 ⋅ 0.1 μ = 1.7 ∑ x 2 ⋅ p ( x ) = 0 2 ⋅ 0.2 + 1 2 ⋅ 0.3 + 2 2 ⋅ 0.2 + 3 2 ⋅ 0.2 + 4 2 ⋅ 0.1 ∑ x 2 ⋅ p ( x ) = 4.5 σ 2 = ∑ x 2 ⋅ p ( x ) − μ 2 = 4.5 − 1. 7 2 = 1.61
σ = 1.61 = 1.268 \sigma=\sqrt{1.61}=1.268 σ = 1.61 = 1.268
2.
X P(X)
0 0.004
1 0.435
2 0.355
3 0.206
μ = 0 ⋅ 0.004 + 1 ⋅ 0.435 + 2 ⋅ 0.355 + 3 ⋅ 0.206 μ = 1.763 ∑ x 2 ⋅ p ( x ) = 0 2 ⋅ 0.004 + 1 2 ⋅ 0.435 + 2 2 ⋅ 0.355 + 3 2 ⋅ 0.206 ∑ x 2 ⋅ p ( x ) = 3.709 σ 2 = ∑ x 2 ⋅ p ( x ) − μ 2 = 3.709 − 1.76 3 2 = 0.6008 \begin{aligned}
&\mu=0 \cdot 0.004+1 \cdot 0.435+2 \cdot 0.355+3 \cdot 0.206 \\
&\mu=1.763 \\
&\sum x^{2} \cdot p(x)=0^{2} \cdot 0.004+1^{2} \cdot 0.435+2^{2} \cdot 0.355+3^{2} \cdot 0.206 \\
&\sum x^{2} \cdot p(x)=3.709 \\
&\sigma^{2}=\sum x^{2} \cdot p(x)-\mu^{2}=3.709-1.763^{2}=0.6008
\end{aligned} μ = 0 ⋅ 0.004 + 1 ⋅ 0.435 + 2 ⋅ 0.355 + 3 ⋅ 0.206 μ = 1.763 ∑ x 2 ⋅ p ( x ) = 0 2 ⋅ 0.004 + 1 2 ⋅ 0.435 + 2 2 ⋅ 0.355 + 3 2 ⋅ 0.206 ∑ x 2 ⋅ p ( x ) = 3.709 σ 2 = ∑ x 2 ⋅ p ( x ) − μ 2 = 3.709 − 1.76 3 2 = 0.6008
σ = 0.6008 = 0.775 \sigma=\sqrt{0.6008}=0.775 σ = 0.6008 = 0.775
3.
X P(X)
1 0.1
2 0.1
3 0.6
8 0.13
19 0.07
μ = 1 ⋅ 0.1 + 2 ⋅ 0.1 + 3 ⋅ 0.6 + 8 ⋅ 0.13 + 19 ⋅ 0.07 μ = 4.47 ∑ x 2 ⋅ p ( x ) = 1 2 ⋅ 0.1 + 2 2 ⋅ 0.1 + 3 2 ⋅ 0.6 + 8 2 ⋅ 0.13 + 1 9 2 ⋅ 0.07 ∑ x 2 ⋅ p ( x ) = 39.49 σ 2 = ∑ x 2 ⋅ p ( x ) − μ 2 = 39.49 − 4.4 7 2 = 19.51 \begin{aligned}
&\mu=1 \cdot 0.1+2 \cdot 0.1+3 \cdot 0.6+8 \cdot 0.13+19 \cdot 0.07 \\
&\mu=4.47 \\
&\sum x^{2} \cdot p(x)=1^{2} \cdot 0.1+2^{2} \cdot 0.1+3^{2} \cdot 0.6+8^{2} \cdot 0.13+19^{2} \cdot 0.07 \\
&\sum x^{2} \cdot p(x)=39.49 \\
&\sigma^{2}=\sum x^{2} \cdot p(x)-\mu^{2}=39.49-4.47^{2}=19.51 \\
\end{aligned} μ = 1 ⋅ 0.1 + 2 ⋅ 0.1 + 3 ⋅ 0.6 + 8 ⋅ 0.13 + 19 ⋅ 0.07 μ = 4.47 ∑ x 2 ⋅ p ( x ) = 1 2 ⋅ 0.1 + 2 2 ⋅ 0.1 + 3 2 ⋅ 0.6 + 8 2 ⋅ 0.13 + 1 9 2 ⋅ 0.07 ∑ x 2 ⋅ p ( x ) = 39.49 σ 2 = ∑ x 2 ⋅ p ( x ) − μ 2 = 39.49 − 4.4 7 2 = 19.51
σ = 19.51 = 4.417 \sigma=\sqrt{19.51}=4.417 σ = 19.51 = 4.417
4.
X P(X)
6 0.1
8 0.11
9 0.61
10 0.09
11 0.09
μ = 6 ⋅ 0.1 + 8 ⋅ 0.11 + 9 ⋅ 0.61 + 10 ⋅ 0.09 + 11 ⋅ 0.09 μ = 8.86 ∑ x 2 ⋅ p ( x ) = 6 2 ⋅ 0.1 + 8 2 ⋅ 0.11 + 9 2 ⋅ 0.61 + 1 0 2 ⋅ 0.09 + 1 1 2 ⋅ 0.09 ∑ x 2 ⋅ p ( x ) = 79.94 σ 2 = ∑ x 2 ⋅ p ( x ) − μ 2 = 79.94 − 8.8 6 2 = 1.44 \begin{aligned}
&\mu=6 \cdot 0.1+8 \cdot 0.11+9 \cdot 0.61+10 \cdot 0.09+11 \cdot 0.09 \\
&\mu=8.86 \\
&\sum x^{2} \cdot p(x)=6^{2} \cdot 0.1+8^{2} \cdot 0.11+9^{2} \cdot 0.61+10^{2} \cdot 0.09+11^{2} \cdot 0.09 \\
&\sum x^{2} \cdot p(x)=79.94 \\
&\sigma^{2}=\sum x^{2} \cdot p(x)-\mu^{2}=79.94-8.86^{2}=1.44
\end{aligned} μ = 6 ⋅ 0.1 + 8 ⋅ 0.11 + 9 ⋅ 0.61 + 10 ⋅ 0.09 + 11 ⋅ 0.09 μ = 8.86 ∑ x 2 ⋅ p ( x ) = 6 2 ⋅ 0.1 + 8 2 ⋅ 0.11 + 9 2 ⋅ 0.61 + 1 0 2 ⋅ 0.09 + 1 1 2 ⋅ 0.09 ∑ x 2 ⋅ p ( x ) = 79.94 σ 2 = ∑ x 2 ⋅ p ( x ) − μ 2 = 79.94 − 8.8 6 2 = 1.44
σ = 1.44 = 1.2 \sigma=\sqrt{1.44}=1.2 σ = 1.44 = 1.2
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