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### Probability Series

Basic Probability Concepts

Conditional Probability and Bayes’ Rule

Discrete Random Variables

Continuous Random Variables

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### Continuous Random Variables

A continuous random variable is a function that maps the sample space of a random experiment to an interval in real value space. A random variable is called continuous if there is an underlying function f(x) such that

• Where f(x) is a non negative function called probability density function (pdf)

• Probability Density funtion can be considered analogous to Probability Mass function of Discrete random variable but differs in that pdf does give probability directly at a value like in case of pmf. Hence the rules of probability do not apply to f(x).

• Pdf takes value 0 for values outside range(X).

• Also the following property can be inferred from rules of probability,

• Probability of a continuous random variable taking a range of values is given by the area under the curve of f(x) for that range.

### Cumulative Distribution Function (cdf)

Cdf for continuous random variable is same as the one for discrete random variable. It is given by,

• Properties of cdf:
• Unlike f(x), cdf(x) is probability and follows the laws and hence,
• As probability is non-negative, cdf is a non-decreasing function.
• Differential of cdf is given by
• Limits of cdf are given by

### Some Specific Distributions

• Uniform Distribution

$X = Uniform(N)$ is used to model a scenario where all outcomes are equally likely. Uniform([c, d]) is when all the values of $x(c \leq x \leq d)$ are equally probable. It is given by

• Exponential Distribution

• Defined using a parameter $\lambda$ and has the pdf given by
• It is used to model the waiting time for an event to occur eg. waiting time for nuclear decay of radioactive isotope distributed exponentially and $\lambda$ is known as the half life of the isotope.

• This distribution exhibits lack of memory i.e. when waiting time is modeled using exponential distributions, the probability of it happening in next N minutes remains same irrespective of the time passed.

• Proof: According to lack of memory property, prove $P(X \gt n+w | X \gt w ) = P(X \gt n)$.

• Normal Distribution
• Most commonly used distribution
• Also known as Gaussian distribution
• It is denoted by $N(\mu, \sigma^2))$ where $\mu$ is the mean and $\sigma^2$ is the variance fo the given distribution.
• Standard Normal Distribution, denoted by Z is a normal distribution with mean = 0 and variance = 1.
• It is symmetric about the y-axis and follows the bell-curve.
• It is given by

### Summary of Distributions

Distribution pdf(x) cdf(x)
$Uni(c, d)$ ${1 \over d-c}$ ${x-c \over d-c}$
$Exp( \lambda ), \, x \geq 0$ $\lambda e^{- \lambda x}$ $1 - e^{- \lambda x}$
$N(\mu, \sigma^2)$ ${1 \over \sigma \sqrt{2 \pi} } e^{ \frac {- (x - \mu)^2} {2 \sigma^2} }$ ${1 \over 2}[1 + erf({ x - \mu \over \sigma \sqrt{2} })]$

### Expected Value

• Gives the average or the mean value over all the possible outcomes of the variable.
• Used to measure the centrality of a random variable.

• For a continuous random variable X, whose pdf is $f(x)$, the expected value in the interval [c, d] is given by,
• Expected value is often denoted by $\mu$.

• $f(x)dx$ denotes the probability value with which X can take the infinitesimal range dx.

• Some other properties of expected value of a random variable:

$E(X+Y) = E(X) + E(Y) \tag{3}$ $E(cX + d) = c * E(X) + d \tag{4}$

### Variance and Standard Deviation

• For a continuous random variable X, with Expected value $\mu$, variance is given by,

$Var(X) = E((X-\mu)^2) \tag{5}$ $\sigma = \sqrt (Var(X)) \tag{6}$

• Some other properties of variance of a random variable:

$Var(X) = E(X^2) - (E(X))^2 \tag{7}$ $Var(aX + b) = a^2 Var(X) \tag{8}$ $Var(X+Y) = Var(X) + Var(Y) \text { iff X and Y are independent } \tag{9}$

### Quartiles

The value of $x$ for which $cdf(x) = p$ is called $p^{th}$ quartile of X. So, median for the continuous random variable is the $0.5^{th}$ quartile