Help with Statistical Inference

[trousmate]

View attachment 3490

See image.

I have to present this on Monday and don't really know what I'm doing having missed a couple of lectures through illness. Any help or hints would be very much appreciated.

Thanks,
TM

 

[chisigma]

https://www.physicsforums.com/attachments/3490

See image.

I have to present this on Monday and don't really know what I'm doing having missed a couple of lectures through illness. Any help or hints would be very much appreciated.

Thanks,
TM

Wellcome on MHB trousmate!...

If You have rsamples of a r.v. Y then the likelihood function is... $\displaystyle f(y_{1}, y_{2}, ..., y_{r} | p) = p^{\sum y_{i}}\ (1-p)^{r - \sum y_{i}}\ (1)$... so that... $\displaystyle \ln f= \sum_{i} y_{i}\ \ln p - (r - \sum_{i} y_{i} )\ \ln (1 - p) \implies\frac{d \ln f}{d p } = \frac{\sum_{i} y_{i}} {p} - \frac {(r -\sum_{i} y_{i})}{1 - p}\ (2)$... and imposing$\displaystyle \frac{d \ln f}{d p} = 0$ You arrive to write... $\displaystyle \overline {p}= \frac{\sum_{i} y_{i}}{r}\ (3)$Kind regards$\chi$ $\sigma$

 

[zzephod]

https://www.physicsforums.com/attachments/3490

See image.

I have to present this on Monday and don't really know what I'm doing having missed a couple of lectures through illness. Any help or hints would be very much appreciated.

Thanks,
TM

For this problem you have the likelihood:

$$L(\theta|Y) = b(Y,r,\theta)=\frac{r!}{Y!(r-Y)!}\theta^Y(1-\theta)^{r-Y}$$

Then the log-likelihood is:

$$LL(\theta|Y) =\log(r!) - \log(Y!) -\log((r-Y)!)+Y\log(\theta) +(r-Y)\log(1-\theta)$$

Then to find the value of $\theta$ that maximises the log-likelihood we take the partial derivative with respect to $\theta$ and equate that to zero:

$$\frac{\partial}{\partial \theta}LL(\theta|Y)=\frac{Y}{\theta}-\frac{r-Y}{1-\theta}
$$
So for the maximum (log-)likelihood estimator $\hat{\theta}$ we have:
$$\frac{Y}{\hat{\theta}}-\frac{r-Y}{1-\hat{\theta}}=0$$

which can be rearranged to give:$$\hat{\theta}(r-Y)=(1-\hat{\theta})Y$$ or $$\hat{\theta}=\frac{Y}{r}$$

and as you should be aware of; the maximum log-likelihood estimator for this sort of problem is also the maximum likelihood estimator.

.

 

[zzephod]

Wellcome on MHB trousmate!...

If You have rsamples of a r.v. Y then the likelihood function is... $\displaystyle f(y_{1}, y_{2}, ..., y_{r} | p) = p^{\sum y_{i}}\ (1-p)^{r - \sum y_{i}}\ (1)$... so that... $\displaystyle \ln f= \sum_{i} y_{i}\ \ln p - (r - \sum_{i} y_{i} )\ \ln (1 - p) \implies\frac{d \ln f}{d p } = \frac{\sum_{i} y_{i}} {p} - \frac {(r -\sum_{i} y_{i})}{1 - p}\ (2)$... and imposing$\displaystyle \frac{d \ln f}{d p} = 0$ You arrive to write... $\displaystyle \overline {p}= \frac{\sum_{i} y_{i}}{r}\ (3)$Kind regards$\chi$ $\sigma$

Very nice but this is the solution to the wrong problem, it just happens to have the same solution as the problem as asked, but it is still the wrong problem. The data is the number of successes from r trials not a vector of results.

.

 

[blue_raver22]

Hello TM,

I'm sorry to hear that you were unable to attend some lectures due to illness. Statistical inference can be a complex topic, but I'm happy to help provide some guidance.

Firstly, statistical inference is the process of drawing conclusions or making predictions about a population based on a sample of data. This is important because it allows us to make generalizations and predictions about a larger group without having to collect data from every single individual.

Some key concepts in statistical inference include hypothesis testing, confidence intervals, and p-values. Hypothesis testing involves setting up a null hypothesis and an alternative hypothesis and then using data to determine whether the null hypothesis can be rejected or not. This helps us determine if there is a significant difference or relationship between variables.

Confidence intervals are a range of values that we can be reasonably confident the true population parameter falls within. They are often used to estimate the true value of a population parameter, such as the mean or proportion.

P-values are a measure of the strength of evidence against the null hypothesis. A smaller p-value indicates stronger evidence against the null hypothesis, and may lead to rejecting the null hypothesis in favor of the alternative.

When presenting on statistical inference, it's important to clearly define the research question and explain the methods used to analyze the data. It can also be helpful to provide visual representations, such as graphs or charts, to illustrate the results.

I hope this helps give you a better understanding of statistical inference. If you have any specific questions or need further clarification, please don't hesitate to reach out. Good luck with your presentation on Monday!

Best,