ProbabilityMassFunction.md (2073B)
1 # Probability Mass Function (PMF) 2 3 L4 4 5 **Definition:** A PMF describes the probability of some mapping of a [RandomVariables](RandomVariables.md) from inputs to a specific output. 6 7 **Important:** PMFs are for discrete random variables whereas PDFs are for continuous. 8 9 This can be displayed as some form of bar graph. 10 11 To find the PMF value for a given point we sum the probability of each input that maps to the output in question. 12 13 ## Example 14 15 ```mermaid 16 17 graph LR 18 a --> x 19 b --> y 20 c --> y 21 d --> z 22 ``` 23 24 In the below example assume each connection is the function defined by the Random Variable. As such, the PMF output for x would be P(a). The PMF output for y would be P(b) + P( c ) and the output for z would be P(d). 25 26 With proper notation (and assuming random variable X) we can state the above as $P_X(x) = P(A), \space P_X(y) = P(B)+P(C)$ etc. 27 28 ## Expected Value 29 30 The expected value of a PMF is the most probable output. This is calculated by summing the probabilities of each output multiplied by the output value. This will be the 'middle' of the sample space. 31 32 This is denoted by the function E[] where the inside is the random variable that is being predicted upon. 33 34 ## Geometric 35 36 The geometric PMF is a specific PMF where every subsequent output decreases by a given percent each time creating a form of poisson distribution. 37 38 The geometric PMF is **memoryless** in that regardless of the step you start on, the future probabilities will be the same because all conditionals result in no added information thus they are independent probabilities. 39 40 ## Conditional (L6) 41 42 Conditional PMFs are just PMFs but they have a specified even that occurred. In these instances we simply resize the sample space accordingly and then recalculate probabilities. 43 44 ## Joint (L6) 45 46 See [JointProbability](JointProbability.md) for joint PMF information. 47 48 ## Marginal (L6) 49 50 The marginal PMF of X is the P_X(x) that can be found from the joint probability of P_{X,Y}(x,y). Basically, we reverse engineer the probability of a given outcome given the sum of the joint probabilities.