In Exercises 79 and 80, (a) show that the nonnegative function is a probability density

Chapter 8, Problem 79

(choose chapter or problem)

Probability A nonnegative function f is called a probability density function if

\(\int_{-\infty}^{\infty} f(t) d t=1\).

The probability that x lies between a and b is given by

\(P(a \leq x \leq b)=\int_{a}^{b} f(t) d t\).

The expected value of x is given by

\(E(x)=\int_{-\infty}^{\infty} t f(t) d t\).

In Exercises 79 and 80,(a) show that the nonnegative function is a probability density function,(b) find \(P(0 \leq x \leq 4)\), and (c) find E(x).

\(f(t)=\left\{\begin{array}{ll} \frac{1}{7} e^{-t / 7}, & t \geq 0 \\ 0, & t<0 \end{array}\right. \)

Text Transcription:

Int^infinity_-infinity f(t)dt=1

P(a leq x leq b)=Int^b_a f(t)dt

E(x)=Int^infinity_-infinity t f(t)dt

P(0 leq x leq 4)

f(t)={1/7 e^-t/7, t geq 0_0, t < 0