The value of the standard deviation is σ = 2.20. Using probability distribution, the required standard deviation is calculated.
How to calculate the standard deviation?
The formula for the standard deviation of the given probability distribution is
σ = √∑([tex]x_i^2[/tex] × [tex]P(X_i)[/tex]) - μₓ²
Where the mean μₓ = ∑[[tex]x_i[/tex] × [tex]P(X_i)[/tex]]
Calculation:
It is given that,
x: $2, $3, $5, $10
P(X=x): 0.55, 0.26, 0.11, 0.08
Step 1: Calculating the mean:
we have μₓ = ∑[[tex]x_i[/tex] × [tex]P(X_i)[/tex]]
⇒ μₓ = 2 × 0.55 + 3 × 0.26 + 5 × 0.11 + 10 × 0.08
∴ μₓ = 3.23
Step 2: Calculating the standard deviation:
x: 2, 3, 5, 10
x²: 4, 9, 25, 100
P(X=x): 0.55, 0.26, 0.11, 0.08
([tex]x_i^2[/tex]) × [tex]P(X_i)[/tex]: 4 × 0.55 = 2.2; 9 × 0.26 = 2.34; 25 × 0.11 = 2.75; 100 × 0.08 =8
∑[([tex]x_i^2[/tex]) × [tex]P(X_i)[/tex]]: 2.2 + 2.34 + 2.75 + 8 = 15.29
Therefore,
The standard deviation, σ = √∑([tex]x_i^2[/tex] × [tex]P(X_i)[/tex]) - μₓ²
⇒ σ = [tex]\sqrt{15.29-(3.23)^2}[/tex]
= [tex]\sqrt{15.29-10.43}[/tex]
∴ σ = 2.20
Learn more about the probability distribution here:
https://brainly.com/question/18804692
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