Probability Function §

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Def: §

Discrete Random Variables

Discrete random varaibles §

Defination: If the set of all possible values of a random variable, X, is a countable set then X is called a discrete random variable. The function If it is clear from the context that X is discrete, then we simply will say discrete pdfProbability Density Funciton and cumulative distribution function(CDF). Another common terminology is Probability Mass Function

if X is a discrete random variable wiuth pdf(x), then the expected valueof X is defined by

Important: Bernoulli Random Variable

Binomial distribution

Probability Mass Function

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For a discrete X that takes value {x}, the probability mass function (p.m.f) must satisfy:

3. Example: Roll a Die twice and let x be the larger outcome. Q: give the distribution of X(all the possible values) |give the p.m.f of X(probability of each value)

or ,

• The p.m.f of x is

X=x&1 &2 &3 &4 &5 &6 \ \frac{1}{36}& \frac{3}{36} & \frac{5}{36} &\dots & & \frac{11}{36} \end{bmatrix}$$ or for x = 1,2,3,4,5,6

ex(Q4 (a)): p.m.f f(x) = x = 1.,2,3,4 find C.

cdf of a discrete X is

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expected value

Discrete Random Variables: For a discrete random variable, the expected value, usually denoted as μ or E(X), is calculated using The formula means that we multiply each value,x, in the support by its respective probability, f(x), and then add them all together. it can be seen as an average value but weighted by the likelihood of the value.

##Example:

x	0	1	2	3	4
f(x)	1/5	1/5	1/5	1/5	1/5

=2

Properties:

Continuous Random Variables: The expected value(or mean) for a continuos random variables , usually denoted as μ or E(X), is calculated using:

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(or population mean)

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Discrete Random variable:

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Key words: §

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TAGS §

#probability#randomvariable