# Topic 4 - Random Variables and Discrete Probability Distributions

## Contents

Gerald Keller (2011), Statistics for Management and Economics (Abbreviated), 9th Edition, pp. 217-243.

## Random Variables & Probability Distributions

• X = Random Variable (RV) - A function or rule that assigns a number to an outcome of an experiment
• Discrete RV – A countable number of values - time, weight or height
• Continuous RV – Values are uncountable - the rugby, soccer or basketball score
• Probability Distribution – Assigns a probability to the values of the RV P(X=x)

### Discrete Probability Distributions

• Requirements
• 0 ≤ P(X=x) ≥ 1 (All probabilities must be between 0 and 1)
• ∑ P(X=x) = 1 (All probabilities must sum to 1)
• Population Mean
• Population Variance

### Laws of Expected Value

• E(c) = c
• E(X + c) = E(X) + c
• E(c*X) = c*E(X)
• E(X + Y) = E(X) + E(Y)

### Laws of Expected Variance

• V(c) = 0
• V (X + c) = V(X)
• V(c*X) = c^2 V(x)
• V(X + Y) = V(X) + V(Y) – 2COV(X,Y)

## Bivariate Distributions

[2] Covariance

This bi-variate distributions shows the join probabilities of X and Y. For example, the probability of X = 2 and Y = 1 is 0.03. The marginal probabilities can be found in the margins. E.g. the P(X=2) = 0.06 + 0.03 + 0.01 = 0.1

## End

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## References

Textbook refers to Gerald Keller (2011), Statistics for Management and Economics (Abbreviated), 9th Edition,.

1. Textbook Pg. 217-228
2. Textbook Pg. 229-242