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4.3 Distributions

Distributions
Discrete
Random variables
Probability
Binomial
Variance
Probability mass function
Continuous
Probability density function
Expectation
Normal distribution
z score
Mean
Standard deviation

Discrete Distributions

If XX is a random variable with sample space {x1,x2,...,xnx_1, x_2, ..., x_n} and corresponding probabilities pip_i, then {p1,p2,...,pnp_1, p_2, ..., p_n} describes the probability distribution of XX. We call P(x)=P(X=x)P(x) = P(X=x) as the probability mass function (PMF).
Distribution
Features
Binomial Distribution
This distribution is used for a binomial experiment with nn independent trials. We can predict rr successes with the probability of success pp. We write X∼B(n,p)X \sim B(n, p).
E(X)=npE(X) = np
Var(X)=np(1−p)Var(X) =np(1-p)
σ(X)=Var(X)\sigma (X)=\sqrt {Var(X)}
Probability mass function: P(X=r)=(nr)pr(1−p)n−rP(X=r)=\binom nr p^r (1-p)^{n-r}
Use binomcdfbinomcdf for specific range and binompdfbinompdf for a specific value.
Note that inclusive (≤, ≥) and exclusive (<, >) matters.

Continuous distributions

A random variable XX is continuous if its cumulative distribution function is continuous. If we have: (HL)
FX(x)=∑−∞xfX(t)dtF_X(x)=\sum _{-∞}^xf_X(t) dt
fXf_X is the probability density function (PDF) of XX. Note that PDF and PMF are fundamentally different objects, as PDFs can be greater than 1. Below, for simplicity, we denote PDF as f(x)f(x).
One example of such a distribution is the normal distribution. (SL)
Distributions
Features
Normal Distribution
We use normal distribution to model continuous random variables. We write X∼N(µ,σ2)X \sim N(µ, \sigma^2) where it draws a bell-shaped curve. The standard normal distribution is X∼Z(0,1)X \sim Z(0, 1).
Use normcdfnormcdf for specific range and normpdfnormpdf for a specific value. Note that inclusive (≤, ≥) and exclusive (<, >) does not matter.
We call z=x−µσz = \frac {x - µ}{\sigma} as the z scorez ~score and is used to estimate the number of standard deviations from the mean. If we know the probability but the mean or the standard deviation, always convert to z scorez ~score first.
Any linear combination of independent normally distributed random variables is itself a normally distributed random variable.
To find t in P(X<t)=cP(X < t)= c with X∼N(µ,σ2)X \sim N(µ, \sigma^2), use invnorminvnorm.
*Note that the calculator only recognizes less than.
4.3 Distributions
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4.3 Distributions