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4 years ago

Calculate the sum of the infinite series

Mathematics

1 answer:

Rina8888 [55]4 years ago

4 0

Answer:

$S_\infty=250$ .

Step-by-step explanation:

The given infinite series is $50+40+32+\frac{128}{5}+...$ .

The first term of this series is $a_1=50$ .

The common ratio is $r=\frac{40}{50}$ .

$\Rightarrow r=\frac{4}{5}$ .

The sum of this infinite series is given by the formula,

$S_\infty=\frac{a_1}{1-r}$

We now substitute all the above values in to this formula obtain,

$S_\infty=\frac{50}{1-\frac{4}{5}}$

This implies that,

$S_\infty=\frac{50}{\frac{1}{5}}$

This simplifies to,

$S_\infty=50\times 5$

$S_\infty=250$ .

The correct answer is C.

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The correlation coefficient is 0.53

<h3>How to calculate the correlation coefficient</h3>

The correlation coefficient (r) is calculated as:

$r = \frac{n(\sum xy) - \sum x \sum y}{\sqrt{[n \sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2}}$

Using the given parameters, we have:

$r = \frac{20 *8249 - 1257* 116}{\sqrt{[20 * 98823 - 1257^2][20 * 836 - 116^2}}$

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$r = \frac{20 *8249 - 1257* 116}{\sqrt{[20 * 98823 - 1580049][20 * 836 - 13456}}$

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$r = \frac{19168}{\sqrt{[396411*3264}}$

Evaluate the product

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2 years ago

5.2.14. For the negative binomial pdf p (k; p, r) = k+r−1 (1 − p)kpr, find the maximum likelihood k estimator for p if r is know

Volgvan

Answer:

$\hat p = \frac{r}{\bar x +r}$

Step-by-step explanation:

A negative binomial random variable "is the number X of repeated trials to produce r successes in a negative binomial experiment. The probability distribution of a negative binomial random variable is called a negative binomial distribution, this distribution is known as the Pascal distribution".

And the probability mass function is given by:

$P(X=x) = (x+r-1 C k)p^r (1-p)^{x}$

Where r represent the number successes after the k failures and p is the probability of a success on any given trial.

Solution to the problem

For this case the likehoof function is given by:

$L(\theta , x_i) = \prod_{i=1}^n f(\theta ,x_i)$

If we replace the mass function we got:

$L(p, x_i) = \prod_{i=1}^n (x_i +r-1 C k) p^r (1-p)^{x_i}$

When we take the derivate of the likehood function we got:

$l(p,x_i) = \sum_{i=1}^n [log (x_i +r-1 C k) + r log(p) + x_i log(1-p)]$

And in order to estimate the likehood estimator for p we need to take the derivate from the last expression and we got:

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And we can separete the sum and we got:

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Now we need to find the critical point setting equal to zero this derivate and we got:

$\frac{dl(p,x_i)}{dp} = \sum_{i=1}^n \frac{r}{p} -\sum_{i=1}^n \frac{x_i}{1-p}=0$

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For the left and right part of the expression we just have this using the properties for a sum and taking in count that p is a fixed value:

$\frac{nr}{p}= \frac{\sum_{i=1}^n x_i}{1-p}$

Now we need to solve the value of $\hat p$ from the last equation like this:

$nr(1-p) = p \sum_{i=1}^n x_i$

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