Question

How does your web browser get a file from the Internet? Your computer sends a request for the file to a web server, and the web server sends back a response. Let $Y=$the amount of time (in seconds) after the start of an hour at which a randomly selected request is received by a particular web server. The probability distribution of $Y$can be modeled by a uniform density curve on the interval from $0\mathrm{to}3600$seconds. Define the random variable $W=Y/60$.

a. Explain what $W$represents.

b. What probability distribution does $W$have?

Short answer

a. $W$denotes the amount of time (in minutes) that a randomly selected request is received by a given web server following the start of an hour.

b. $W$exhibits a consistent density curve from data-custom-editor="chemistry" $0\mathrm{to}60$minutes

Step-by-step solution

Part(a) Step 1 : Given Information   

Given:

The probability distribution of $Y$can be modeled by a uniform density curve on the interval from : $0\mathrm{to}3600$ seconds.

Random variable : $W=Y/60$.

Part(a) Step 2 : Simplification 

$Y:$The amount of time (in seconds) that a randomly selected request is received by a certain web server following the start of an hour.

So, $W=\frac{Y}{60}$

We know that $Y$indicates the amount of time (in seconds) that a randomly selected request is received by a particular web server following the start of an hour.

We also know that a minute is divided into $60$seconds.

If we divide the time in seconds by $60$, we get the time in minutes.

As a result, $W$denotes the amount of time (in minutes) that a randomly selected request is received by a given web server following the start of an hour.

Part(b) Step 1 : Given Information   

Given :

The probability distribution of $Y$can be modeled by a uniform density curve on the interval from : role="math" localid="1654002781486" $0\mathrm{to}3600$seconds.

Random variable : $W=Y/60$.

Part(b) Step 2 : Simplification 

$Y$has a uniform density curve in the range of $0\mathrm{to}3600$seconds for probability distribution.

So, $W=\frac{Y}{60}$

We know that $Y$indicates the amount of time (in seconds) that a randomly selected request is received by a particular web server following the start of an hour.

$Y$also has a consistent density curve that spans the time range of data-custom-editor="chemistry" $0\mathrm{to}3600$seconds.

Because $W=60$, each data value in the Y distribution is split by the same constant.

If every data value is divided by the same constant, the form of the distribution remains intact.

As a result, the shape of $W\text{'}s$distribution is the same as that of $Y\text{'}s$.

This means that $W\text{'}s$ density curve is uniform

$0$seconds equals zero minutes for a homogeneous density curve.

$3600\mathrm{seconds}\mathrm{equals}60$minutes in the same way.