# Probability Distributions - Some Concepts

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1. Discrete and continuous random variables;

__Discrete random variable__ can take on at most a countable number of possible values, such as coin flip or rolling dice.

__Continuous random variable__ can take on an uncountable (infinite) number of possible values, such as asset returns or temperatures.

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**2. R**ange of possible outcomes of a specified random variable;

1. Coin toss

{Head, Tail}.

2. Rolling a die

(1,2,3,4,5,6}.

3. Share returns (in percent)

Interval [-100, +∞ ).

3. Probability distribution;

__Probability distribution__ specifies the probabilities of the possible outcomes of a random variable.

4. Probability function

__Probability function__ specifies the probability that the random variable takes on a specific value: *P(X = x)*. To determine if a given function is a probability function it must fulfill the two key properties in the next LOS.

5. **P**roperties of a probability function;

__Two Key Properties of a Probability Function__.

*i. **0 ≤ p(x) ≤ 1* because a probability lies between 0 and 1.

*ii. *The sum of probabilities *p(x)* over all values of *X* equals 1.

6. ** C**umulative distribution function

__Cumulative Distribution function__ specifies the probability that the random variable X is less than or equal to a particular value *x, P(X ≤ x)*. For a discrete random variable this is the sum of the probabilities for all values less than or equal to *x*.

7. Probability density function;

__Probability Density function (pdf)__ specifies the probability that a continuous random variable takes on a specific value.

*i. *Pdf of a number is a function .

*ii.*Integral over the range of the r.v. equals 1, .

8. **D**iscrete uniform random variable

__Discrete Uniform Random Variable:__ The uniform random variable X takes on a finite number of values, *k*, and each value has the same probability of occurring, i.e. *P(x _{i}) = 1/k *for

*i = 1,2,…,k*.

Examples of simple uniform random variables:

*i. *__Coin flip__: Prob(head) = ½ Prob(tail) = ½

*ii. *__Die__: Prob(Die shows 1) = 1/6, Prob(Die shows 2) = 1/6, …Prob(Die shows 6)=1/6

9. ** **binomial random variable and binomial probability distribution;

__Bernoulli random variable__ is a binary variable that takes on one of two values, usually 1 for success or 0 for failure. Think of a single coin flip as an example of a Bernoulli r.v.

__Binomial random variable:__

*X ~ B(n, p)*is defined is the number of successes in

*n*Bernoulli random trials where

*p*is the probability of success on any one Bernoulli trial. The probability distribution for a Binomial random variable is given by:

p(x|n,p) = ^{n}C_{x} px(1-p)^{n-x}

The distribution is symmetric when *p = .5*, but otherwise it is skewed.

9A. ** E**xpected value and variance of a binomial random variable;

__Binomial__: E(X)= np

*X ~ B(n,p)*Expected Value__Binomial__: σ² = np(1-np)

*X ~ B(n,p)*Variance**10. K**ey properties of the normal distribution;

__Normal distribution__ is a continuous, symmetric probability distribution that is completely described by two parameters: its mean, μ, and its variance, σ^{2}. Written as *N(μ, σ ^{2}).*

*i. *The normal distribution is said to be bell-shaped with the mean showing its central location and the variance showing its “spread”.

*ii. *A linear combination of two or more Normal random variables is also normally distributed.

Very good explanation of binomial probability with example.In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli distribution.

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