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

Here are questions 33 and 34 to assist in answering question 52.
33) Let X be a random variable with probabilitydensity
f(x)={c(1-x^2), -1therwise
(a) What is the value of c?
(b) What is thecumulativedistributionfunction of X?
34) Let theprobability

Let X be a random variable with probabilitydensity
f(x) = c(1-x^2), if -1therwise
Determine the value of the constant , and findthecumulativedistributionfunction of X

1) Suppose that A, B, C are independent random variables, each being uniformly distributed over (0,1). What is the joint cumulativedistributionfunction of A, B, C? What is theprobability that all of the roots of the equation Ax^2+Bx+C=0 are real?
2) If X is uniformly distributed over (0,1) and Y is exponentially distribute

I)Let X be an exponential random variable with mean 1,and Y = exp^(X/2):
a)find F(y)
b)Evaluate E(Y)
c)Evaluate E[(Y^2)/(1+(X^2))]
II)Therandom variable X is uniformly distributed on the interval [1,3].Findtheprobabilitydensityfunction fy(y)of therandom variable Y=2X+5

Let X denote a continuous random variable with probability density function f(x) = kx^3/15 for 1≤ X ≤ 2.
a. Determine the value of the constant k.
b. Determine the probability that X > 1.5.
c. Determine thecumulativedistributionfunction F(x) and state the values of F(x) at x = 0.5, 1.5, and 2.5.

4. The probability density function if X, the lifetime of a certain type of electronic device (measured in hours} is given by:
f(x) =
10/x^2 for x>10
and
=0 for x<=10
(a) Find P {X > 20}
(b) What is thecumulativedistributionfunction of X?
(c) What is theprobability that of 6 such types of devices at least 3 will

ProbabilityDensityfunction, Probabilitydistribution, and Probability
Could someone give me definitions with examples of each.
Please make the explanations as clear as possible.

34. Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked as an exponential random variable with parameter 1/20. Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is theprobability that she would get at least 2

Consider the following exponential probability density function:
F(x) = (1/14)e^(-x/14) for x>=0
This number represents the time between arrivals of customers at the drive-up window of a bank.
a. Find f(x<=7)
b. Find f(3.5<=x<=7)