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grigory [225]
3 years ago
15

g Consider the experiment of a single roll of an honest die and a single toss of 3 fair coins. Let X be the value on the die and

let Y be the number of heads obtained on the coins. Find the joint probability function of X and Y
Mathematics
1 answer:
klemol [59]3 years ago
4 0

Answer:

The probability function of X and Y is

P(X = k, Y = 0)  = 1/48\\P(X = k, Y = 1) = 1/16\\P(X = k, Y = 2) = 1/16\\P(X = k, Y = 3) = 1/48

With k in {1,2,3,4,5,6}

Step-by-step explanation:

We can naturally assume that X and Y are independent. Because of that, P(X=a, Y=b) = P(X=a) * P(Y=b) for any a, b.

Note that, since the die is honest, then P(X=k) = 1/6 for any k in {1,2,3,4,5,6}. We can conclude as a consequence that P(X=k, Y=l) = P(Y=l)/6 for any k in {1,2,3,4,5,6}.

Y has a binomial distribution, with parameters n = 3, p = 1/2. Y has range {0,1,2,3}. Lets compute the probability mass function of Y:

P_Y(0) = {3 \choose 0} * 0.5^3 = 1/8

P_Y(1) = {3 \choose 1} * 0.5* 0.5^2 = 3/8

P_Y(2) = {3 \choose 2} * 0.5^2*0.5 = 3/8

P_Y(3) = {3 \choose 3} * 0.5^3 = 1/8

Thus, we can conclude that the joint probability function is given by the following formula

P(X = k, Y = 0) = 1/8 * 1/6 = 1/48\\P(X = k, Y = 1) = 3/8 * 1/6 = 1/16\\P(X = k, Y = 2) = 3/8 * 1/6 = 1/16\\P(X = k, Y = 3) = 1/8 * 1/6 = 1/48

For any k in {0,1,2,3,4,5,6}

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Equation  1):

You want to buy something that costs $166.66, and it's on sale for 25% off. What is the item's sale price?

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So for this sale, you'll save $41.67 on this item.

This means, the cost of the item to you is  

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Now what's 75.000% of the total cost?  

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Equation 2:)

You want to buy something that costs $125, and it's on sale for 25% off. What is the item's sale price?

First, convert the 25% to a real mathematical number. For percent's, this is always done by dividing the 25% by 100%, or 25% / 100% = 0.250.

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0.250 x $125 = $31.25.

So for this sale, you'll save $31.25 on this item.

This means, the cost of the item to you is  

$125 - $31.25 = $93.75.  

Alternatively, you can think about it this way. The item is 25% off. This means you'll pay 75.000% of the total cost (100% - 25% = 75.000%).

Now what's 75.000% of the total cost?

0.750 x $125 = $93.75.

Answer =====> Just like the result above, the sale price on the item is $93.75.




Hope that helps!!!                                                     : )


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