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Featured 8 months ago

Since the events are independent,

Featured 6 months ago

It is a term used to denote applying the maximum likelihood approach along with a log transformation on the equation to simplify the equation.

For example suppose i am given a data set

If

Here the equation is the same but the paramter of interest is

However before doing so I see that I can apply the natural log before finding the derivative to solve for

so an approximation of

Using MLE we can also find out what the estimated standard deviation is.

Featured 2 months ago

We use our good friend Bayes' rule to help us deduce:

**Warning: Long answer ahead!**

One form of Bayes' rule states the following:

This allows us to write a conditional probability of "B given A" in terms of "A given B", which may be easier to calculate. In this question,

B = "die rolled twice", and

A = "score is 5".

Let's write this into the equation:

The numerator on the RHS is easy to deduce; however, the denominator, in its current state, is not. But we can make it easier. We recall that, if

We actually have that here; the

Kind of a "the whole is equal to the sum of its parts" thing.

From hereon, let's use the shorthand

Now, we do the calculations!

Finally, we place these values back into the equation for

For completeness, *had* to take either 1, 2, or 3 rolls.

There's a great Numberphile video on YouTube discussing Bayes' rule here:

Featured 1 month ago

See below:

We'll start with burger patties on top. Burgers with cheese will be in

#color(white)(0)1color(white)(00000000000000)2color(white)(0000000000000)3color(white)(00000)"Burger patties"#

#77%color(white)(0000000000)16%color(white)(00000000000)7%#

#/color(white)(000)"\"color(white)(0000000000)"/"color(white)(00)"\"color(white)(0000000000)"/"color(white)(000)"\"#

Ok - how to do this:

We know there are 3 types of burger options: 1 patty, 2 patties, 3 patties and these will add to 100%:

We know that

We know that

We know that 54% of all orders have cheese. We know the total orders for 1 and 3 patties, so we can solve for 2 patties:

And lastly we can figure out the number of 2 patties no cheese:

Featured 1 week ago

A hypothesis is what informs an experiment or what is being tested/measured. It is often called an educated guess.

A hypothesis is what informs an experiment or what is being tested/measured. It is often called an educated guess.

Examples of hypotheses:

- As the number of cigarettes a person smokes per day increases, the risk of lung cancer increases.
- Black cats never get adopted from animal shelters.

A hypothesis can be further broken down into a **null hypothesis** and an **alternative hypothesis**. The null hypothesis states that there is no relation and the alternative hypothesis states that there is a relation.

Referring to the first example given above, the null hypothesis would be that there is no relationship between the number of cigarettes a person smokes per day and the risk of lung cancer. The alternative hypothesis is that there is an effect of the number of cigarettes a person smokes per day and the risk of lung cancer OR that the larger the number of cigarettes a person smokes per day, the larger the risk of lung cancer.

A good hypothesis should not only be clear and informative, but it also needs to be measurable.

Hypotheses should be developed after studying the problem or issue as thoroughly as possible, building upon previous knowledge and observations.

Featured 1 week ago

Mean:

Variance:

Standard deviation:

We are given that

The question is, when we roll this die once, what value should we expect to get? Or perhaps, if we roll the die a huge number of times, what should the average value of all those rolls be?

Well, of the 100% of the rolls, 15% should be "0", 35% should be "1", 45% should be "2", and 5% should be "3". If we add all these together, we'll have what's known as a *weighted average*.

In fact, if we placed these relative weights at their matching points on a number line, the point that would "balance the scale" is the mean that we seek.

This is a good way to interpret the mean of a discrete random variable. Mathematically, the mean

#mu = E[X] = sum_("allÂ " x)[x * P(X=x)]#

In our case, this works out to be:

#mu = [0*P(0)]+[1*P(1)]+[2*P(2)]+[3*P(3)]#

#color(white)mu=(0)(0.15)+(1)(0.35)+(2)(0.45)+(3)(0.05)#

#color(white)mu="Â Â Â Â Â Â Â "0"Â Â Â Â Â Â Â "+"Â Â Â Â "0.35"Â Â Â Â "+"Â Â Â Â Â "0.9"Â Â Â Â Â "+"Â Â Â Â "0.15#

#color(white)mu=1.4#

So, over a large number of rolls, we would expect the average roll value to be

The variance is a measure of the "spread" of

That's because the variance *squared* distance between each possible value and

#sigma^2="Var"(X)=E[(X-mu)^2]#

Using a bit of algebra and probability theory, this becomes

#sigma^2=E[X^2]-mu^2#

#color(white)(sigma^2)=sum_("allÂ x")x^2P(X=x)" "-" "mu^2#

For this problem, we get

#sigma^2=[0^2*P(0)]+[1^2*P(1)]+[2^2*P(2)]#

#color(white)(sigma^2=)+[3^2*P(3)]" "-" "1.4^2#

#color(white)(sigma^2)=(0)(0.15)+(1)(0.35)+(4)(0.45)+(9)(0.05)#

#color(white)(sigma^2=)-1.96#

#color(white)(sigma^2)=0.64#

So the average squared distance between each possible

Standard deviation is easyâ€”it's just the square root of the variance. But, why bother with it if it's pretty much the same? Because the units of *square* of the units of

That's where standard deviation comes in. The standard deviation

#sigma= sqrt (sigma^2)#

For this problem, that works out to be

#sigma = sqrt(0.64)=0.8#

So every time we pick an *(See: confidence intervals.)*

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