Answer:
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Step-by-step explanation:
The fundamentals
A continuous random variable can take infinite values in the range associated function of that variable. Consider
is a function of a continuous random variable within the range
, then the total probability in the range of the function is defined as:
![\int\limits_a^b {f\left( x \right)dx} = 1 a∫b f(x)dx=1](https://tex.z-dn.net/?f=%5Cint%5Climits_a%5Eb%20%7Bf%5Cleft%28%20x%20%5Cright%29dx%7D%20%3D%201%20a%E2%88%ABb%E2%80%8B%09%20f%28x%29dx%3D1)
The probability of the function
is always greater than 0. The cumulative distribution function is defined as:
![F\left( x \right) = P\left( {X \le x} \right)F(x)=P(X≤x)](https://tex.z-dn.net/?f=F%5Cleft%28%20x%20%5Cright%29%20%3D%20P%5Cleft%28%20%7BX%20%5Cle%20x%7D%20%5Cright%29F%28x%29%3DP%28X%E2%89%A4x%29)
The cumulative distribution function for the random variable X has the property,
![0 \le F\left( x \right) \le 10≤F(x)≤1](https://tex.z-dn.net/?f=0%20%5Cle%20F%5Cleft%28%20x%20%5Cright%29%20%5Cle%2010%E2%89%A4F%28x%29%E2%89%A41)
The probability density function for the random variable X has the properties,
![\\\begin{array}{c}\\{\rm{ }}f\left( x \right) \ge 0\\\\\int\limits_{ - \infty }^\infty {f\left( x \right)dx} = 1\\\\P\left( E \right) = \int\limits_E {f\left( x \right)dx} \\\end{array} f(x)≥0](https://tex.z-dn.net/?f=%5C%5C%5Cbegin%7Barray%7D%7Bc%7D%5C%5C%7B%5Crm%7B%20%7D%7Df%5Cleft%28%20x%20%5Cright%29%20%5Cge%200%5C%5C%5C%5C%5Cint%5Climits_%7B%20-%20%5Cinfty%20%7D%5E%5Cinfty%20%7Bf%5Cleft%28%20x%20%5Cright%29dx%7D%20%3D%201%5C%5C%5C%5CP%5Cleft%28%20E%20%5Cright%29%20%3D%20%5Cint%5Climits_E%20%7Bf%5Cleft%28%20x%20%5Cright%29dx%7D%20%5C%5C%5Cend%7Barray%7D%20f%28x%29%E2%89%A50)
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