Question

If a random variable X has a cumulative distribution function F(x) given by

0 , x≤0
F(x) = c (x-e^(-x)) , 0<x<1
5 , x≥1


then find its corresponding probability distribution function and hence calculate
P(0<x<1)

Answer 1

Answer on Question #51467 – Math – Statistics and Probability

If a random variable $X$ has a cumulative distribution function $F(x)$ given by

$$0, x \leq 0$$ $$F(x) = c (x - e^{\wedge}(-x)), \quad 0 < x < 1$$ $$1, x \geq 1$$

then find its corresponding probability distribution function and hence calculate

$$P(0 < x < 1)$$

Solution.

There was a mistake in assignment, because $F(x)$ is related to probability, but probability is not greater than 1.

A cumulative distribution function of $X$ is given by

$$F(x) = \begin{cases} 0, & x \leq 0 \\ c (x - e^{-x}), & 0 < x < 1 \\ 1, & x \geq 1 \end{cases}$$ $$\begin{array}{l} \text{Probability distribution function } p(x) = \frac{dF}{dx} = \begin{cases} 0, & x \leq 0 \\ c (1 + e^{-x}), & 0 < x < 1 \\ 0, & x \geq 1 \end{cases} \\ P(0 < X < 1) = P(X < 1) - P(X \leq 0) = F(1) - \lim_{x \to 0^+} F(x) = 1 - c. \end{array}$$

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