Continuous Distribution

1. Continuous Distribution: Sample space is infinite, and We cannot record the frequency of each distinct value. Thus we can no longer represent these distributions with a table what we can do is represent them with a graph more precisely the graph of the probability density function or PDF short. The associated probability for every possible value “y”. f(y) >= 0
2. The greater the denominator becomes the closer the fraction is to 0.
3. The probability for any individual value equal to 0. P(X) = 0
4. Cumulative Distribution Function (CDF): this function encompasses everything up to a certain value. We denote the CTF as capital F of Y for any continuous random varibale Y as the name suggests it represents probability of the random variable being lower than or equal to a specific value since no value could be lower than or equal to negative infinity. The CDF of value for negative infinity would equal zero similarly since any value would be lower than plus infinity. We would get a 1 if we plug infinity into the distribution function. Discrete distributions also have DFS but they are used far less frequently. That is because we can always add up to PDF values associated with the individual probabilities.
5. CTF is especially useful when we want to estimate the probability of some interval. Graphically the area under the density curve would represent the chance of getting a value within that interval we find this area by computing the integral of the density curve over the interval from a to b.
6. The integral for the PDF to CDF. The derivative for the CDF to PDF.
7. The CDF represents the sum of all the PDF values up to that point.
8. Using a capital letter N followed by the mean and variance of the distribution we read the following notation as variable X follows a normal distribution with mean mu and variance sigma squared when dealing with actual data. We would usually know the numerical values of mu and sigma squared.
9. The expected value for a normal distribution equals its mean. Variance sigma squared is usually given when we define the distribution.
10. The variance of a variable is equal to the expected value of the squared variable minus the squared expected value of the variable. 68, 95, 99.7 law
11. The graph of the Normal Distribution possess the Bell-shaped, Symmetric and Thin tails.
12. Transformation: A way in which we can alter every element of a distribution to get a new distribution with similar characteristics. For normal distributions, we can use addition subtraction, multiplication and division without changing the type of the distribution.
13. Standardizing is a special kind of transformation in which we make the expected value equal to zero and the variance equal to one the distribution we get after standardizing any normal distribution is called a standard normal distribution.
14. Standardizing is incredibly useful when we have a normal distribution. However, we cannot always anticipate that the data is spread out that way a crucial factor remeber about the normal distribution is that it requires a lot of data. If our sample is limited we run the risk of outlines drastically affecting our analysis in cases where we have less than 30 entries.
15. Use the t to define a student’s t distribution followed by a single parameter in parantheses called degrees of freedom.
16. Application for the student’s t distribution: Frequently used when conducting statistical analysis, hypothesis testing with limited data and CDF table (T - table).
17. The application for the Chi-Squared Distribution: Few events in real life, Statistical analysis (Hypothesis testing, computing confidence intervals) and Goodness of fit.
18. The graph of the Chi-Squared distribution is asymmetric, heavily-skewed to the right and has a fat tail on the right, and no tail on the left.
19. The PDF of such a function would start off very high and sharply decrease within the first few timeframes. The curve somewhat resembles a boomerang with each handle lining up with the x and y axis.
20. Rate parameter: how fast the CDF/PDF curve reaches the point of plateauing and how spread out the graph.
21. The PDF plateaus around the 0 mark, since the more y increases, the less likely it is to attain such a value, where as the CDF always has to reach the 1 mark.
22. The logistic distribution is defined by two key features. It’s mean and its scale parameter. The former dictates the center of the graph whilst the latter shows how spread out the graph is going to be.
23. The 3 features for the CDF of a logistic distribution:
    [1] It follows an S-shape and plateaus around the 1 value.
    [2] The probability drastically starts to increase once we reach values close to the mean.
    [3] The steeper the curve is, the faster it reaches value close to absolute certainty (1).