Problem

A health-food store stocks two different brands o…

Problem 156 Easy Difficulty

A restaurant serves three fixed-price dinners costing $\$ 20, \$ 25,$ and $\$ 30 .$ For a randomly selected couple dining at this restaurant, let $X=$ the cost of the man's dinner and $Y=$ the cost of the
woman's dinner. The joint pmf of $X$ and $Y$ is given in the following table:
(a) Compute the marginal pmfs of $X$ and $Y .$
(b) What is the probability that the man's and the woman's dinner cost at most $\$ 25$ each?
(c) Are $X$ and $Y$ independent? Justify your answer.
(d) What is the expected total cost of the dinner for the two people?
(e) Suppose that when a couple opens fortune cookies at the conclusion of the meal, they find
the message "You will receive as a refund the difference between the cost of the more
expensive and the less expensive meal that you have chosen." How much does the
restaurant expect to refund?

Answer

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Intro Stats / AP Statistics

Probability with Applications in Engineering, Science, and Technology

Chapter 4

Joint Probability Distributions and Their Applications

Section 1

Jointly Distributed Random Variables

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Video Transcript

So this question we have been given that this Annie's arrival Damn analogies that every time some digging access, uh, Chinese. But every time I'm getting why, as, uh, Ivy's But I have a dime. Okay? And if the older that accent by so uniformly distributed on the double five coma six. Okay, so to start with the first part, what they've lost, we've been given the joint pdf off x and y. And based on that, we're supposed to write down the, uh, the marginal pdf sand the joint BDs. So here I can say that of the joint BDF pdf can be off laying eggs and why you can run down this as productive off marginal. Pdf ce off Ex sent by And this were saying because both x and y are independent evened. So since they're independent, I'm going to use this formula and forced right down the marginal PDS. So first I'm writing for next. So we safe effects is since the uniform distribution does one and zero. So just one between the double 5 to 6 as forgiven question and it is zero and sweat, and similarly, for my also this will be one and zeal for five between, but I also between five and six, and it is zero otherwise so, combining them, we can ride out the joint pdf that therefore, the joint pdf is if off x comma y can be denies. This is one for the conditions off excess exit between five and six and, uh, similarly, why is also between five and six and Tzeitel answer So this week Britain is the joint belief of X and bite. Next part is too cancel. It would be when we say the probability the board arrived between 5 15 and 5 45 So to find this probability, we got to write this in terms of X and buying doubles. So which represents as excess between 5 15 on 5 45 Similarly, they have I also between fine 15 and 5 45 Now, since the diamond over of you converting this into a number so we can convert Ah 5 15 to 1 number by, uh, taking this as 5 15 minutes, five hours so into edo, 60 minutes to convert it into minutes. And plus this is 15 minutes, right, and we can divide descent or two by 16 minutes to again converted into a number. So which actually gives me 5.25 And in a similar way, 5.4 fire also lettuce converts. This is five hours into 60 minutes. Plus we have another 45 minutes, and the whole thing is a 1,000,000 divided by 60 minutes. So in terms of numbers, this will now change to five point 75 Which means that indirectly, now this probability, which I'm trying to find I can write this as remember Eddie or for using the same product or swooned over the multiplication of individual except by values. But the values are changing from 5.255 point 75 So we take. This is 5.25 less and equal to X. There's an equal to 5.75 and in tow. Probability off 5.25 Lesson equal. Vilas unequalled If I 0.75 So if I multiply these two values, then huh I asked for the definition. The integral for X and by will be given by India. Elation off one because the function is one but the limiters from five point through fire to 5.75 and into the next part is integration off one device with the same limbs? Yeah. So if I indicate, then we're getting the value here as X from 5.25 to 5.75 and in tow. The next is by 5.255 point 75 So we get this value as 0.5 into a 0.5 and so you're 0.25 So this is the heartbeat, and we've just completed now to draw the part. See, I'm Wolf was considered or diagram where we can find out the other details from the diagram. So let me just say, like the dying, I'm forced. And it is just be sciences. Diarrhea? Yeah, So you can see. Then we've drawn two lines which represents the line. First is why minus X is equal one by 60 others X minus y is equal one basics along with the coordinate points that we have brought here and we have shaded the triangle regions also based on the values that are given to us. So now from this diagram, you can see that the points are taken in minutes and writing here that the points, huh? Taken in minutes originally. Okay, but in equation, when we're solving it has to be an hour. So I'm saying, but an equation, it is in powers. So because of this now I'm going to find out the required probability. So we say therefore, the require probably can we agree. Written as this notation we have given, the question is more off. X minus way is less than equal one by six you saw in the graph of plotting these equations and now you can see that this can be given by one minus probability off would the triangle So which is Triangle ABC And the other is ah, strangled the f So this will be one minus provocative. ABC will be the values that we have that it's half in tow. The other value is fine. By six into five by six. This actually represents the idiot who would happen to base into height minus here. It'll to be 1/2 and do five by six into five by six Simplify! I'm getting this answer as you 0.30 Firefight