**an individual who has automobile insurance from a certain company is randomly selected** This is a topic that many people are looking for. **savegooglewave.com** is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, ** savegooglewave.com ** would like to introduce to you **Expected Values : Solved Example #1**. Following along are instructions in the video below:

This lecture. We are going to solve these. Two questions so lets start with with question.

Number. 1. An individual who has automobile insurance from a certain company is selected let vy be the number of moving violations for which the individual was cited during the last three years.

The probability mass function of y. Is given here in part. A we have to compute the expected value of y.

So in part a we have to compute the expected value of y. We know that expected value of y is equal to summation y multiplied by probability of y. So this is equal to 0 multiplied.

By zero point six zero plus. 1. Multiplied by zero point.

Two five plus. 2. Multiplied.

By zero point. One zero plus. 3.

Multiplied by zero point zero. 5. And this is equal to zero.

Plus zero. Point. Two five plus zero point two zero plus zero point 1 5.

And this is equal to zero point six. So the expected value of y is zero point six. Now lets move to part b suppose.

An individual with violations incurs a surcharge of 100 y square. We have to calculate the expected amount of the surcharge. So that means we have to calculate the expected value of 100 y square.

Now we know that in general expected value of a x is equal to ay multiplied by expected value of x where x is any kind of random variable. We are given in the question that weiser random variable. So that even y square is a random variable and using this proposition.

That i have written here. We can write expected value of 100 y square is equal to 100 multiplied by expected value of y square. So now.

Lets calculate the expected value of y square. So. Lets calculate this so to calculate the expected value of y square.

Lets calculate the probability mass function of y square. So we are given in the equation. The probability mass function of y.

And we have to calculate the probability mass function of y square. You can write square here. So we are given that y can take these 4 values and the.

Corresponding. Probabilities. Are 06 025 01 and 005.

Now when y is equal to 0. Y. Square is also equal to.

0 and the probability that y will be equal to. 0 is 06 so the probability that y square will be equal to. 0 is also 06.

When y is equal to 1 y. Square is also equal to 1 and the probability that y is equal. To 1 is.

025 so even this probability is 025. When y is equal to 2 y square is equal to. 4 and this would have a probability of 01.

And when y is equal to 3 y square is equal to. 9 and once. Again we have probability of 005.

Here so now as you can see the probability values in these cases are identical. And thats logical as well okay. So now lets calculate the expected value of y square.

So expected value of y square is equal to summation of y square. X. You can write either probability of y square.

Or probability of y. So lets write probability of 5. It doesnt make any difference.

Both are same and we are summing it over all the values of. Y square so we can write 0. Into.

06. Plus. 1.

Into. 025. Plus.

4 multiplied. By. 01.

Plus. 9. X.

005 and this is. Equal. To.

0. Plus 025. Plus.

04 plus. 045. And this is equal to 1 point.

1. So expected value of y square. Is equal to 1 point.

1. And expected value of 100 y square is equal to 100 multiplied by expected value of y square. So.

This means that expected value of 100 y square is equal to 100 multiplied by 1 point. 1. And this is equal to 110 dollars.

And thats it this is the answer to part b now lets move to question number. 2 an appliance dealer sells three different models of upright freezers having 135 fifteen point nine and nineteen point one cubic feet of storage space. Respectively let x.

Is equal to the amount of storage space purchased by the next customer to buy a freezer. Suppose that x has the following pmf in part a we have to compute the expected value of x expected value of x square and variance of x. So in part a lets compute.

The expected value of x first. And we know that expected value of x is equal to summation x multiplied by probability of. X so this is equal to thirteen point five multiplied by.

02. Plus fifteen point nine multiplied by 05. Plus nineteen point one multiplied by zero point three and this is equal to sixteen point three eight so expected value of x is equal to sixteen point three eight lets now calculate the expected value of x square so expected value of x square is equal to summation x square multiplied by probability of.

X okay and this is equal to 135 whole square multiplied by 02. Plus. Fifteen point nine whole square multiplied by zero point five plus nineteen point one whole square multiplied by zero point.

Three. I am solving this we will get two seventy two point two nine eight. So this is the expected value of x square.

Now the third thing that we have to find is the variance of x. There are many formulas to find the variance of x. But as we have already calculated the expected value of x.

And the expected value of x square. So lets use this formula. So it is equal to expected value of x square.

Minus. Expected value of x whole square. Okay.

And we have already calculated these values. So this is equal to two seventy two point two nine eight minus sixteen point three eight whole square. And this is equal to three point nine nine three six.

So with this we are done with the part a lets now move to part b. If the price of a freezer having capacity eggs cubic feet is twenty five x minus eight point five. What is the expected price paid by the next customer to buy a freezer.

So this means that in part b. We have to calculate the expected price so that means we have to calculate this because the price is equal to 25 x. Minus eight point five.

Now we can use the propositions that we know about the expected values. So. We know that expected value of a y plus b.

Where. Y. Is any random variable is equal to a x.

Expected. Value of y plus b. So.

We can use this proposition. To calculate the expected value of price. So.

Using this proposition. We get expected value of 25 x. Minus.

8. Point. 5.

Is equal to 25 multiplied by expected value of x minus. 8. Point.

5. And in the part a. We calculated the expected value of x.

It was equal to sixteen point three eight so 25 multiplied by sixteen point three eight minus eight point five. And this is in a bracket and this is equal to four zero nine point five minus eight point five. And this is equal to four zero one so the expected price paid by the next customer to buy a freezer is four zero one.

Now. Lets move to part c. In part c.

We have to calculate the variance of the price paid by the next customer. So that means we have to calculate variance of 25 x. Minus.

Eight point five well in general. We have this proposition that variance of a vy plus b. Where y is any random variable is equal to a square multiplied by a variance of y.

So we can use this proposition to answer. This question. So using this proposition variance of 25 x.

Minus. Eight point five is equal to 25 whole square multiplied by variance of x. And we calculated the variance of x in part a of this question.

It was equal to three point nine nine three six. So this is equal to 625 x. Three point nine nine three six and this is equal to 2 4.

9. 6. So this is the variance of the price.

Now lets move to part d. Suppose. That although the rating capacity of a freezer is.

X the actual capacity is this h x is equal to x minus. 01. X.

Square. What is the expected actual capacity of the freezer purchased by the next customer so that means we have to calculate the expected value of x minus. 01.

X. Square well using the propositions that we have for expected values we can write this as expected value of x minus. 01.

Expected value of x square. And in part a of this question. We calculated that expected value of x.

Is sixteen point. Three eight and the expected value of x square. Is 272 point two nine eight solving this we get thirteen point six five seven and this is the answer.

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