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##### Exam P Images

Chapter 1-General Probability Reading 1 Question 4

##### Exam P Syllabus – Learning Outcomes

General Probability 1.a – Define set functions, Venn diagrams, sample space, and events. Define probability as a set function on a collection of events and state the basic axioms of probability. 1.b – Calculate probabilities using addition and multiplication rules. 1.c – Define…

##### State and apply the Central Limit Theorem

For this learning objective, a certain knowledge of the normal distribution and knowing how to use the Z-table is assumed. The central limit theorem is of the most important results in the probability theory. It states that the sum of…

##### Calculate probabilities and moments for linear combinations of independent random variables

Given random variables $$X_1,\ X_2,\ldots,X_p$$ and constants $$c_1, c_2,\ldots,\ c_p$$ then: $$Y=c_1X_1+c_2X_2+\ldots+c_pX_p$$ is a linear combination of $$X_1,\ X_2,\ldots,\ X_p$$. Mean of a Linear Combination If $$Y=c_1X_1+c_2X_2+\ldots+c_pX_p$$, then: $$E\left(Y\right)=c_1E\left(X_1\right)+c_2E\left(X_2\right)+\ldots+c_pE(X_p)$$ This true because recall that, if we have  $$u\left(X,Y\right)=X+Y$$, and let’s say…

##### Determine the distribution of a transformation of jointly distributed random variables

Transformation for Bivariate Discrete Random Variables Let $$X_1$$ and $$X_2$$ be a discrete random variables with joint probability mass function $$f_{X_1,X_2}(x_1,x_2)$$ defined on a two dimensional set $$A$$. Define the following functions: $$y_1 =g_1 (x_1, x_2)$$ and  $$y_2 =g_2(x_1,x_2)$$…

##### Explain and apply joint moment generating functions

We can derive moments of most distributions by evaluating probability functions by integrating or summing values as necessary. However, moment generating functions present a relatively simpler approach to obtaining moments. Univariate Random Variables In the univariate case, the moment generating…

##### Calculate moments for joint, conditional, and marginal random variables

Moments of a Probability Mass function The n-th moment about the origin of a random variable is the expected value of its n-th power. Moments about the origin are $$E(X),E({ X }^{ 2 }),E({ X }^{ 3 }),E({ X }^{ 4 }),….\quad$$ For…

##### Determine conditional and marginal probability functions

Conditional Distributions Conditional probability is a key part of Baye’s theorem, which describes the probability of an event based on prior knowledge of conditions that might be related to the event. It differs from joint probability, which does not rely…

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