What is the prime factorization of 45/72

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:

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

And we can separete the sum and we got:

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

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

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

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

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:

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

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

$p \sum_{i=1}^n x_i +nrp = nr$

$p[\sum_{i=1}^n x_i +nr]= nr$

And if we solve for $\hat p$ we got:

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

And if we divide numerator and denominator by n we got:

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

Since $\bar x = \frac{\sum_{i=1}^n x_i}{n}$

4 0

3 years ago

Allison can jog 3 blocks in the same amount of time it takes her brother, Blake, to jog 2 blocks. If they keep jogging at the sa

Nitella [24]

Answer:

Blake will jog 12 blocks

Step-by-step explanation:

Allison : Blake = 3 : 2

If they keep jogging at the same rate, how many blocks will Blake jog if Allison jogs 18 blocks?

Allison : Blake = 18 : x

Equate the ratios

3 : 2 = 18 : x

3/2 = 18/x

Cross multiply

3 * x = 2 * 18

3x = 36

Divide both sides by 3

x = 36 / 3

= 12

x = 12

Blake will jog 12 blocks

6 0

3 years ago

Read 2 more answers

I need help with underlined problems.

vaieri [72.5K]

#7 is 0.04 and it is a terminating decimal

4 0

3 years ago

A butcher has 10 3/4 pounds of ground beef that will be priced at 3.40 dollars per pound. He divides the meat into 8 equal packa

spin [16.1K]

Well we know that 10 and 3/4 is equivalent to 10.75. If we divide the meat into 8 equal packages we have to divide 10.75 by 8. Dividing 10.75 by 8 gives us 1.34375 pounds of meat per package. If we take that, and multiply it by our given value of $3.40 per pound, we end up getting $4.57 a package.

8 0

3 years ago

Read 2 more answers

From class we learned that in 2010 the federal debt (in millions) was $13,561,623 and the population of the U.S. was 309 million

Crank

According to these numbers, in 2010 the federal debt per capita was $43,888.75. Hope this helps!

6 0

3 years ago

Read 2 more answers