Probability

Question 1

Give the formula for

1) Conditional Probability
2) to check if a variable is independent to the other

Answer 1

P(A|B) = P(A^B) / P(B)  a different way to write it is:
P(A|B) = P(A,B) / P(B)

When E and F are independent, you can check that this gives: P(A|B] = P(A) which is the mathematical way of expressing that knowing F occurred gives us no
additional information about whether E occurred.

# | means given. So A givens B. So we investigate what happens with A if B is given.
# ^ means that both happen. Note: it is actually written down, not up.

Question 2

Baye´s Theorem.

1. Derive it from a decision tree and state it like it´s normally stated.
2. What notation to use to say: not happens?

Answer 2

1. P(A|B) = P(A)P(B|A) / P(B) # den decision tree Weg entlang wird multipliziert.
2. • | (mittelstrich ist oben, sodass 90 Grad Winkel)

Question 3

Continous Distributions

What do we do with a Probability Density Function (PDF) and what is it?

Answer 3

we represent a continuous distribution with a probability density function (pdf) such that the probability of seeing a value in a certain interval equals the integral of the density function over the
interval

f(x) represents the height of the curve at point x. Imagine the function as a smooth histogram, like a mountain landscape, where the area underneath is filled out.

for continous random variables probabilities are areas unter the curve.

The area under the entire curve is equal to one.

Question 4

What is a cumulative distribution function (cdf)?

Answer 4

While pdf describes the shape of the distribution, cdf describes the area in it (the accumulated probability), from a certain point (p.e. x) to the LEFT.

Question 5

Name the 3 forms of pdf and cdf and their formulas

Answer 5

uniform:
Pdf= f(x)= 1/ b-a
CDF = AL= x-a/b-a #AL is Fläche, a linker Punkt, x mitte and b right

exponential:
Pdf= f(x)= 入e- 入x # 入 is highest point, * means hoch - 入x
CDF: 1-e- 入x

standars:
other card

Question 6

Describe the Normal Distribution (Gaussian).

Name the formula!

Answer 6

by deault the Standard Deviation σ (sigma) is 1 and the mean μ (mu) is 0.
Mu shows where the bell is centered, sigma the width.
1 SD apart gets 68%, 2 SD gets 95, 3SD gets 99,7

Formula: 1 / σ √2π e(hoch)-(x-μ)2 / 2σ2

Question 7

What is a Bernouille trial?

Answer 7

In a Bernouille trial there is only success and failure. A coin flip is either heads or tails, so only two possible outcomes with the same probability. Every time the trial is conducted, the probabilities are the same.

Question 8

What does a Confidence Interval do?

Answer 8

• CI is a range of values where your assumption should fall in (p.e. if you think the mean will be 4.3 you can p.e. have a range of 2.6-5.9)
• a way to measure how well your sample represents the population you are studying
• The probability that the confidence interval includes the true mean value within a population is called the confidence level of the CI (normally by 95%)

Question 9

What is a p-value?

What is p-hacking?

Answer 9

Tells us if the observations can support the 0-Hypothesis. Is p-value under 0,05 you can reject the 0-Hypothesis, and thus say that the alternative Hypothesis is true. If that happens - throwing 0 away for alternative, it is called statistical significance

Misinterpretation and intentionally pushing the value under 0,05 to increase the chance of getting published is called p-hacking