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Probability Distribution I-Discrete variables (Variance, Var(x) (image …
Probability Distribution I-Discrete variables
Discrete Probability Distribution
Graphically
symmetricalcentral :star: value=expectation=3
non-symmetrical
Functional Form
Table Form
Discrete Random Variables: This is a variable that can only take on particular values (1,2,3). For example, heads and tails of a coin. It is impossible to obtain 1/8 tails.
The total all probability = 1
The probability are between the value of 0 to 1
Mode: x- value that have the highest probability
Expectation: Average or mean
Expectation is the sum of x-values multiplied by the probability
Expectation of any function of X :warning: expectation can be extended to any function
Total probability=1
Variance, Var(x)
If a discrete random variable has
k
possible values X1,X2,....X
k
, the probabilities would be p1,p2,....p
k
Population Variance: the set of values is possible outcomes
Two independence random variables
E(X+Y)=E(X)+E(Y)
E(aX+bY)=aE(X)+bE(X)
E(aX+bY)=aE(X)-bE(X)
Var(X+Y)=Var(X)+Var(Y)Var(aX+bY)=a^2Var(X)+b^2Var(Y)
Var(aX+bY)=a^2Var(X)
-
b^2Var(Y)
Distribution of X1+X2...+Xn
Comparison the distribution of X1+X2 and 2X
X1 and X2 is two independent observation while 2X is random variable
Standard deviation
The cumulative distribution function, F(x)
The probability up to certain value are summed to give cumulative probability
