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Two messages are to be sent. The time (min) necessary to send each message has an exponentialdistribution with parameter $\lambda=1,$ and the two times are independent of each other. It costs $\$ 2$ . per minute to send the first message and $\$ 1$ per minute to send the second. Obtain the densityfunction of the total cost of sending the two messages. [Hint: First obtain the cumulativedistribution function of the total cost, which involves integrating the joint pdf.

$f_T(t)= e^{t^2} - e^{- t}$ for $ t > 0$

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 1

Jointly Distributed Random Variables

Probability Topics

The Normal Distribution

Missouri State University

Piedmont College

Cairn University

Lectures

0:00

06:48

Ann is expected at $7 : 00…

03:15

Find the cumulative distri…

10:17

Consider the pdf for total…

05:14

The joint density function…

02:58

Verify that $f$ gives a jo…

23:51

Suppose $ X $ and $ Y $ ar…

01:28

08:21

The completion time $X$ fo…

01:13

Determine the cumulative d…

02:15

05:41

Let $X$ be a nonnegative c…

06:37

Commuting to work requires…

05:51

Consider a random sample $…

00:33

Use the following informat…

03:27

05:06

Find the expected value, t…

05:55

01:03

01:21

Determine which are probab…

07:23

We're told that two messages are decent, the time necessary to send each message as an exponential distribution with parameters Lambda One and the two times are independent of each other. We're told that it costs $2 per minutes and first message and $1 per minute send the second message were asked to find the density function of the total cost of sending the two messages So we'll let X and wide the transmission times for these messages. Then it follows that the joints probability density function of X and Y is f X Y. Because they're independent is going to be the marginal probability density fx of x times f y of why you know that both of these are exponential with parameter lambda equals one. So this is going to be each of the negative x times each of the negative y, which is the same a z e to the negative X plus y, and this is valid for X and y greater than zero now will define Iran invariable t to be two times X plus Why, and this represents the total cost to send the two messages you want to find first the cumulative distribution function of t. So we have that f t of tea. This is going to be probability that t is less than or equal to t some given time. T. Marcus It's cost sorry, which is the same as the probability that's two x plus y is less than or equal to t, which is the same as the probability that why is less than or equal to t minus two X Now, of course, if X is such that it's greater than t over to and we have that t minus two X is going to be less than zero. So it follows that this probability zero, because why must be greater than zero? On the other hand, yeah. If X is less than or equal to t over to probability that why is less than a quarter to over mhm t minus two x. This is going to be since the CDF, the integral of the pdf. So we're going to integrate first over the possible values of X so zero to t over to and then over the possible values of why we have that Why can range from zero up to T minus two X And then we calculated earlier that f of x y was either the negative X plus y and then d Y d x. This is a pretty easy, integral solve. Um, you could even break this up into a product of integral is using through Beanies. The're, um But after doing so, you obtain that the integral is one minus two each of the negative tier two plus e to the negative t, and this is only valid for t greater than zero. That's the CDF of tea and therefore the pdf of tea is the derivative of the CDF 50 which, using the expression we just found this is going to be e to the negative t over to minus e to the negative t and this is only valid for t greater than zero.

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