Using the probability density function to find the cumulative distribution function of an exponentially distributed random variable.

You are told that the continuous random variable X is exponentially distributed with parameter a (a > 0). A standard result then says that the probability density function of X is (see attachment)

f(x) = aexp(-ax) for x > 0.

Use this to prove that the corresponding cumulative distribution function F(x) (sometimes referred to as the distribtuion function) is

F(x) = 1 - exp(-ax).

Hence find the probability that X is greater than 5, if a=0.5

Denote the cumulative distribution function of a continuous random variable X by F(x). This is related to the probability density function f(x) via the relation (see attachment)

F(x)=P(X < x) = ...

Solution Summary

You are told that the continuous random variable X is exponentially distributed with parameter a (a > 0). A standard result then says that the probability density function of X is (see attachment)

f(x) = aexp(-ax) for x > 0.

Use this to prove that the corresponding cumulative distribution function F(x) (sometimes referred to as the distribtuion function) is

F(x) = 1 - exp(-ax).

Hence find the probability that X is greater than 5, if a=0.5

... the derivative dX/dY must be used by absolute ... of transformations to ﬁnd the probability density function of Y ... the distribution of Y . 6. Use moment generating ...

... Electronic calculators may be used, provided that they cannot ...probability that the next 100 customers using Till 1 ... we're interested in P( X ≥ k ) , we use: ...

... Electronic calculators may be used, provided that they cannot store ... result of E (etX ) , we just use it again ... variance can always be found by using the formulae ...

... about citing these materials or our Terms of Use, visit: http ... and to intro duce the notations that will be used in the ... 2π M! M =0 ( I I. 2 6 ) using the power ...

... plant, the best approach would be to use a. zero ... The minimal-spanning technique would best be used a. by ...Using simulation for a queuing problem a. would be rare ...

... for a new manufacturing plant, the best approach would be to use. ... 33. The minimal-spanning technique would best be used. ...Using simulation for a queuing problem. ...

... The lost income by not using the truly ... 0 or 1 = F or T lime land use, coded landuse distance ... Bayesian methods (eg weights of evidence) are often used in risk ...

... lottery, the revenues from which will be used to fund ... µ X − 60 Standardizing the variable X using Z = = and ... 7 of 9 cumulative X p(X) probability 0 0.00674 ...

... The lost income by not using the truly ... 0 or 1 = F or T lime land use, coded landuse distance ... Bayesian methods (eg weights of evidence) are often used in risk ...