Probability Distribution I-Discrete variables

Discrete Probability Distribution

Graphically

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.

Expectation: Average or mean

Expectation of any function of X ⚠ expectation can be extended to any function

Variance, Var(x)

Two independence random variables

Distribution of X1+X2...+Xn

Comparison the distribution of X1+X2 and 2X

The total all probability = 1

Expectation is the sum of x-values multiplied by the probability

If a discrete random variable has k possible values X1,X2,....Xk, the probabilities would be p1,p2,....pk

Population Variance: the set of values is possible outcomes

Standard deviation

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The probability are between the value of 0 to 1

Mode: x- value that have the highest probability

symmetricalcentral ⭐ value=expectation=3

non-symmetrical

The cumulative distribution function, F(x)

Total probability=1

The probability up to certain value are summed to give cumulative probability

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)

X1 and X2 is two independent observation while 2X is random variable