Question

An urn contains N numbered balls from 1 to N, where N ? 2. A subset of n balls, where n < N, is randomly drawn without replacement from the urn. Let X be the sum of the numbers marked on the n balls drawn from the urn. (a) Determine P(X ? n(n+1)/2 + 1). (b) Determine the expected value of X.

          An urn contains N numbered balls from 1 to N, where N ? 2. A subset of n balls, where n < N, is randomly drawn without replacement from the urn. Let X be the sum of the numbers marked on the n balls drawn from the urn.
(a) Determine P(X ? n(n+1)/2 + 1).
(b) Determine the expected value of X.
        
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An urn contains N numbered balls from 1 to N, where N ? 2. A subset of n balls, where n < N, is randomly drawn without replacement from the urn. Let X be the sum of the numbers marked on the n balls drawn from the urn.
(a) Determine P(X ? n(n+1)/2 + 1).
(b) Determine the expected value of X.

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A First Course in Probability
A First Course in Probability
Sheldon M. Ross 5th Edition
Chapter 7
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An urn contains N numbered balls from 1 to N, N ≥ 2. A subset of n balls, n < N, is randomly drawn without replacement from the urn. Let X be the sum of the numbers marked on the n balls drawn from the urn. (a) Determine P(X ≤ n(n+1)/2 + 1). (b) Determine the expected value of X.
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8) An urn contains N numbered balls from 1 to N, where N ≥ 2. A subset of n balls, where n < N, is randomly drawn without replacement from the urn. Let X be the sum of the numbers marked on the n balls drawn from the urn. (a) Determine P(X ≤ n(n+1)/2 + 1). (b) Determine the expected value of X.

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An urn contains $n$ balls - the ith having weight $W(i), i=1, \ldots, n .$ The balls are removed without replacement one at a time according to the following rule: At each selection, the probability that a given ball in the urn is chosen is equal to its weight divided by the sum of the weights remaining in the urn. For instance, if at some time $i_{1}, \ldots, i_{r}$ is the set of balls remaining in the urn, then the next selection will be $i_{j}$ with probability $W\left(i_{j}\right) \mid \sum_{k=1}^{r} W\left(i_{k}\right)$, $j=1, \ldots, r .$ Compute the expected number of balls that are withdrawn before ball number 1 .

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Balls numbered 1 through $N$ are in an urn. Suppose that $n, n \leq N$, of them are randomly selected without replacement. Let $Y$ denote the largest number selected. (a) Find the probability mass function of $Y$. (b) Derive an expression for $E[Y]$ and then use Fermat's combinatorial identity (see Theoretical Exercise 11 of Chapter 1) to simplify.

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Transcript

-
00:01 It says to sketch a path for the soccer ball.
00:06 Let's start with that and then we can work on the equation later.
00:09 So it travels a horizontal distance of 150 feet.
00:17 Now we know that when you kick a soccer ball it's going to go up and then because of gravity it's going to come back down.
00:27 Something like this at some point.
00:30 Right there.
00:30 And we know that this...
00:32 Let me do that in a different color.
00:37 We know that this horizontal distance here is 150 feet.
00:45 And we also know that it reaches a height of 100 at its highest point.
00:53 So right there that distance is going to be 100 feet...
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