03 Discrete Random Variables
Discrete Random Variables¶
takes finite/countably-infinite no of values
PDF¶
\[ \begin{aligned} f(x) &= P(X = x) \\ f(x) &\ge 0 \\ \sum f(x) &= 1 \end{aligned} \]
CDF¶
\[ \begin{aligned} F(x) &= P(X \le x) \\ &= \sum\limits_0^x f(x) \\ P(a \le X \le b) &= \sum\limits_a^b f(x) \end{aligned} \]
Terms¶
| Notation | Formula | |
|---|---|---|
| \(E(x)\) | \(\mu\) | \(\sum x \cdot f(x)\) |
| \(E(x^2)\) | \(\sum x^2 \cdot f(x)\) | |
| \(V(x)\) | \(\sigma^2\) | \(E(x^2) - [E(x)]^2\) |
| \(\text{SD}(x)\) | \(\sigma\) | \(\sqrt {V(x)}\) |
| Normalised Variable | \(z\) | \(\dfrac{x - E(x)}{\text{SD}}\) |
\[ \begin{aligned} E(k) &= k & E(kx) &= k \cdot E(x) & E(z) &= 0\\ V(k) &= 0 & V(kx) &= k^2 \cdot V(x) & V(z) &= 1 \end{aligned} \]
Distributions¶
| Distribution | \(f(x)\) | \(\mu\) | \(V(x)\) | |
|---|---|---|---|---|
| Bernoulli | - 2 outcomes - independent & identical trial | \(p\) | \(p(1-p)\) | |
| Binomial | \(n\) indepedent Bernoulli events w/ replacement | \(nC_x \cdot p^x \cdot (1-p)^{n-x}\) | \(np\) | \(np(1-p)\) |
| Hypergeometric | \(n\) dependent Bernoulli trials without replacement | \(f(x) = \frac{MC_x \times (N-M) C_{(n-x)} }{NC_n}\) \(\text{max}\Big(0, n- (N-m) \Big) \le x \le \text{min}(n, M)\) | \(n \left(\dfrac M N \right)\) | \(\left( \dfrac{N-n}{N-1} \right) \cdot n \cdot \dfrac M N \left( 1 - \dfrac M N \right)\) |
| Negative Binomial | \(p=\) Probability of success after \((r-1)\) failures | \(f_x(x) = \begin{cases} \begin{pmatrix} x-1\\ r-1 \end{pmatrix} p^r q^{x-r}, & x= r, r+1, \dots \\ 0, & \text{o.w.} \end{cases}\) | \(\dfrac{rq}{p}\) | \(\dfrac{rq}{p^2}\) |
| Geometric | Negative binomial dist with \(r=1\) No of failures before first success | |||
| Poisson | discrete phenomenon in continuous interval Poisson dist can simulate binomial dist with small value of \(p\) | \(\dfrac {e^{-\mu} \times \mu^x}{x!}\) | \(\alpha t\) | \(\alpha t\) |
Rate Parameter \((\alpha)\)¶
occurences per unit interval
\(\alpha = \dfrac 1 \beta\)
(\(\beta\) will be discussed in next topic)