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Alpha sets the standard for how extreme the data must be before we can reject the null hypothesis. The p-value indicates how extreme the data are. We compare the p-value with the alpha to determine whether the observed data are statistically significantly different from the null hypothesis:
-If the p-value is less than or equal to the alpha (p&lt; .05), then we reject the null hypothesis, and we say the result is statistically significant.
- If the p-value is greater than alpha (p &gt; .05), then we fail to reject the null hypothesis, and we say that the result is statistically nonsignificant.

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SBH: What is the difference between level of significance, &alpha; and p-value?

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Significance level is the probability of when the null hypothesis is true.
Alpha set the range of data before we can reject the null hypothesis.
P-value show what is the value the data falls at.

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SBH: Draw a Markov chain diagram for a real-life example. (Probabilities need not be exact)

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A child is playing with a Transformers figure which can transform into 3 modes, "car”, “helicopter” and “robot”. The observation is such that the mode of the Transformers figure is noted at every minute.

Labelling the state space {1 = robot, 2 = car, 3 = helicopter} the transition matrix for this example is

P     =    |    0.9         0.075   0.025     |

|  0.15        0.8      0.05    |
|  0.25        0.25      0.5      |

Inspiration taken from: https://en.wikipedia.org/wiki/Markov_chain#/media/File:Finance_Markov_chain_example_state_space.svg

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Interval (CL, CU) is the confidence interval (CI) such that an intended parameter (mean/ variance/ standard deviation). 95% CI of a mean is having a significance level alpha=5%. Alpha is the critical region or aka the possible error for the mean to be in the interval while (1-alpha)*100 is the confidence level which we confident the mean lies in the particular interval. CI can be applied in real life such as the interval of the weight of 100 students out of all UTP students.

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The confidence interval describes the uncertainty associated with a sampling method. Suppose we used the same sampling method to select different samples and to compute a different interval estimate for each sample. Some interval estimates would include the true population parameter and some would not. A 90% confidence level means that we would expect 90% of the interval estimates to include the population parameter; A 95% confidence level means that 95% of the intervals would include the parameter.For example, “For the European data, one can say with 95% confidence that the true population for wellbeing among those without TVs is between 4.88 and 5.26.” The confidence interval here is “between 4.88 and 5.26“.

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A cl gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.
Example: student measuring the boiling temperature of a certain liquid observes the readings (in degrees Celsius) 102.5, 101.7, 103.1, 100.9, 100.5, and 102.2 on 6 different samples of the liquid. He calculates the sample mean to be 101.82. If he knows that the standard deviation for this procedure is 1.2 degrees, what is the confidence interval for the population mean at a 95% confidence level?

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The 95% confidence interval(CI) of a mean is the population mean has 95% confidence falls in the CI and 5% chances will not falls in the CI. The 5% is distributed into two part which is upper tailed and lower tailed of the standard distribution curve. The CI is estimated with sample size and sample mean.

Besides, no estimate can have 100% reliability to represent the whole population hence 95% CI means the CI has 95% reliability that the population mean will falls in the interval. CI can applied in real life for example, a business might estimate that a machine uses 10 lbs. of plastic for each unit of a product created. Because the machine cannot be expected to use precisely 10 lbs. per unit, a confidence interval can be created to give a range of possibilities. The company might predict that there is a 95 percent chance that the machine uses on average between 9.85 and 10.5 lbs. of plastic per unit. The confidence interval in this example is 95 percent, and the likelihood that the actual amount of plastic used is outside the estimated range is 5 percent.<br><br></div>

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SBH: Describe the 95% confidence interval (CI) of a mean and give an example how CI can be applied in real life.

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Linear transformation affect the normal curve by standardizing any P(X=x) that can be expressed with the equivalent in P(Z=z) & computed by using the standard normal table with the parameters of  mean(u) and standard deviation(σ) for Z is equal to (X-u)/σ  to obtain closed form expression.

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µ is Mean and σ is Standard Deviation.

Linear transformation is transform a distribution with a given mean and standard deviation into another distribution with mean of 0 and standard deviation of 1.0. The normal curve will be transformed into a STANDARD normal curve.
We can use linear transformation to find the area under the curve or probabilities easily by referring to Standard Normal Distribution Table.

Other than that, we also able to compare the values from different distribution. This can be done by expressing the values as Z and refer to Standard Normal Distribution Table to get the standard value then compare.

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SBH: How does linear transformation Z = (X-µ)/σ affect the normal curve and why do we use it?

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Exponential random variable describes the time between or within Poissons random variable which is continuous yet independent at a constant average rate. An example: The average number of customers in an hours is Poisson BUT the amount of time between a customer and the next one is exponential RV. Since exponential RV can take infinite amount of values of time it is considered continuous.

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Exponential rv is measured in time domain, where the number of event between a time interval is fixed to 1, the time intervals vary. Poisson rv is measured in number domain, where the time intervals between events are constant, the number of events vary and discrete. Exponential rv is a continuous rv, because the time domain is continuous, the time intervals between events are uncountable. The possible outcome is infinite. Poisson rv: number of patients queing at 5 min interval. Exponential rv: time interval between 2 patients called to see doctor, one patient at a time
.

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Miss S:  One of the winners. Good representation of the students' dilemma.

A venn diagram about student's preparation before exam and thier results. The venn diagram shows 4 mutually exclusive and exhaustive events..

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SBH: A good observation on the human blood.

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Miss S:  A winning observation. Good!

Deleted the old picture. The old venn diagram is wrong because it is not mutually exhaustive events as the sample space aren't specified.

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Miss S:  One of the winners. Thank you for sharing this info with the rest of the class.

Tonight moon phase either it will be
1. Full moon
2. New moon
3. Crecent
4. Gibbous

The moon will never be in two phase at one time unless the world is ending

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Miss S:  One of the winners

The event of getting Spade, Heart, Club and Diamond when choosing a card is mutually exclusive and exhaustive. This is because the event s arena impossible to happened at the same time.

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Miss S:  One of the winners. Nice diagram, btw.

Probability of Picking A good or bad fruit out of a basket full of pears and apples placed in a refrigerator in Village 5 D.

There are 4 mutually exclusive events which are:
1) good apple
2) bad apple
3) good pear
4) bad pear

It is impossible to pick an apple that is both good and bad. Likewise, it is impossible to pick a good fruit that is both a pear and an

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SBH: Draw a Venn diagram to illustrate a real-life example of 4 mutually exclusive and exhaustive events. Label each event.

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Mutually exclusive/disjoint events are not independent events, because:

it does not fulfill the requirement of P(A|B)=P(AnB)/P(B)=P(A).

For the events to be independent, P(AnB)=P(A)P(B) and not equals to 0 as P(A) and P(B) have (non-zero) values.

In this case, there is no intersection and thus P(AnB)=0, resulting in P(A|B)=0, which is not equal to P(A).

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SBH: Are mutually exclusive / disjoint events, independent events? Give your reason(s).

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SBH: Compare exponential rv with Poisson rv. Why exponential rv is a continuous rv?

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•A continuous random variable is a random variable that has an infinite number of possible values which is the opposite of discrete random variable that can only take certain values. For CRV, it is not necessarily to have every possible number (like from -ve infinity to +ve infinity) but it can be continuous between two values such as 1 and 2 for example. This means CRV could be 1,2 and everything in between 1.00, 1.01, 1.001, 1.001, .....

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1. the value is obtained by measuring, such as the height of EE student
2. probability distribution showed by density curve
3.  The probability that a continuous random variable X is exactly equal to a number is zero
4. takes all value between given intervals of numbers

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SBH: What is continuous random variable?

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•Variance is a measurement of the spreads between numbers in a data set. The variance measures how far each number in the set is from the mean. Variance is calculated by taking the differences between each number in the set and the mean, squaring the differences (to make them positive) and dividing the sum of the squares by the number of values in the set.

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•In a set of data point, a variance is the average of the squared differences from Mean. In a probability distribution, a variance measures the spread, variability and distribution.

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SBH: In your own words, explain what variance tells us about a set of data points? How about variance of a probability distribution?

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•In Order because every order will count as one outcome..

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•Having "in order" will produce larger samples size due to the fact that an arrangement of a combination creates a unique outcome in the sample space. Say we have letters X, Y and Z in the sample space, an outcome with arrangement of XYZ is a different outcome than XZY, YXZ, YZX, ZXY or ZYX. e.com/GIAG2.crlox:mozart-app-template

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SBH: "In order" or "without order": Which condition will produce a larger sample space? Why?

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•When two or more events do not intersect each other in a given sample space (each of the event contain unique/different elements)

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•Mutually exclusive is where multiple events that does not intersect each other in a given sample space

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SBH: Describe mutually exclusive or disjoint events, in your own words.

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•The number of times v6 internet fail to connect

•S={0, 1, 2,3,4,5,6,….}

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•No.of people buys food from food vending machine in one minute interval

•Sample Space = {0,1,2,...}

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•Gender of student being elected as class student representative

•SS =｛male, female}

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SBH: Give an example of an experiment/observation you can conduct in daily life and describe the sample space you can PROBABLY obtain.

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SBH: Post your mind map on continuous probability distributions.

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1) the trials are independent because the values do not affect the value of the other
2) the set of X values is finite or countably infinite as they cannot be uncountable or it would form a continuous function
3) the pmf (F(x)) is a set of probability values assigned to each value of x as they are discrete values, not continuous

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•1. all trials are independent for each type of discrete random variables except for hypergeometric

2. order doesn't matter (combination) for binomial & hypergeometric

3. for binomial & poisson, x=0,1,2,3,... while for geometric, x=1,2,3,...

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1-Binomial Distribution- X is the number of 'success' among n trials
2-Poisson Distribution-the probability of x occurrences in an interval
3-Geometric Distribution-X is number of trials to first success

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•1. Discrete random variable uses the idea of Bernoulli Trials in which an experiment has two possible outcomes either a 'success' or 'failure'.
2. In an experiment, the trials are independent as they do not affect the outcomes with each other.
3. Lastly, the probabilities of both success and failure are constant throughout the experiment.

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SBH: Give 3 important points that you have learnt about identifying discrete random variables. Give your reasons for each.