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Continuous Random Variables

When will that bus finally arrive? How hot is it going to be outside today? These any many other real-world values can be modeled by continuous random variables.

Cumulative Distribution Function

If the cumulative distribution function of a continuous random variable \(X\) is \[F(x)=\begin{cases}0\qquad&(x<0)\\ \frac{1}{50}x^2\qquad&(0\leq x<5)\\ -\frac{1}{50}x^2+\frac{2}{5}x-1\qquad&(5\leq x\leq10)\\ 1\qquad&(x>10) \end{cases}\] what is \(P(4\leq X\leq6)?\)

If the cumulative distribution function of a continuous random variable \(X\) is \[F(x)=ax~(0\leq x\leq 9),\] what is \(P(1\leq X\leq5)?\)

Which of the following represents the graph of the cumulative distribution function of a continuous random variable?

(A)

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(B)

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(C)

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(D)

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If the cumulative distribution function of a continuous random variable \(X\) is \[F(x)=a\sqrt{x-3}~(3<x\leq 5),\] which of the following represents the probability density function \(f(x)?\)

If the probability density function of a continuous random variable \(X\) is \[f(x)=\frac{a}{x+1}~(0\leq x\leq 4),\] which of the following represents the cumulative distribution function \(F(x)?\)

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