Discrete Random Variables
discrete random var definition
- random var taking at most a countable number of values β discrete β DRV
probability mass function
- for a DRV X, p(a) of X:
- positive for at most a countable number of values of a
Expected Value / Expectation: average outcome of a RV
- if x is DRV with β expectation / expected value of X, :
- a game with = 0 is a fair game
- each player has equal chance of winning or losing
- probability of winning = probability of losing
- eg. flip coins, Casino games are NOT a fair game
Proposition
- if X is DRV taking on 1 of values , with respective probabilities , then, for any real-valued function ^484c8dbb-1a4c-10ce
Corollary
- if a and b are const
Variance
- if X is RV with mean , then Var(X) is the expectation of the square of difference between X and its mean
Standard Deviation
\sigma(X) = \sqrt{Var(X)}
Var(X) = E[X^2] - (E[X])
Var(aX+b) = a^2Var(X)
<!--ID: 1708098043655--> ## Bernoulli Random Variables - Suppose a trial whose outcome is either success or failure. If letting X = 1 when success, X = 0 when failure, the probability mass function of X is: $p(0) = P(X=0) = 1-p$ $p(1) = P(X = 1) = p$ with $p, 0 \leq p \leq 1$ is the probability of success trial -> random variable X is Bernoulli Random Variable (BerRV) -> a BerRV is just BinoRV with parameter $(1,p)$ <!--ID: 1708098043658--> ## Binomial Random Variable - $n$ independent trials, each results in success with probability $p$, or failure with probability $1-p$. Denote $X$ as successes occur in the $n$ trials -> $X$ is binomial random variable with parameter $(n,p)$ <!--ID: 1708098043662--> ## Binomial Experiment - same experiment repeated in fixed number of times - only 2 outcomes, success or failure - repeated trials are independent, so that probability of success remans same for each trial <!--ID: 1708098043666--> ## Binomial Probability - If p is probability of success in single trial of a bino experiment, the prob of x successes and n-x failures, in n independent repeated trials of the experiment, known as bino probability - $P(x \text{ successes in } n \text{ trials}) = \binom{n}{x}p^x(1-p)^{n-x}$ - If X is bino RV with para n and p - $E[X] = np$ - $Var(X) = np(1-p)$ <!--ID: 1708098043669--> ## Poisson Random Variable - a RV X taking on one of the values 0,1,2,... is a PRV with para $\lambda$ if, for some $\lambda > 0, i = 0,1,2,\ldots$ - $p(i) = P(X = i) = e^{-\lambda} \frac{\lambda^i}{i!}, \quad i = 0,1,2,\ldots$ - $\lambda$ is average rate of value - This equation defines a probability mass function, since - $\sum_{i=0}^{\infty} p(i) = e^{-\lambda} \cdot \sum_{i=0}^{\infty} \frac{\lambda^i}{i!} = e^{-\lambda}\cdot e^{\lambda} = 1$ - PRV can be used as approx for BinoRV $X \sim B(n,p)$: $n$ is large, $p$ is small, $\lambda = np$ moderate size $X \sim Po(\lambda) -> P(X=i) = e^{-\lambda} \frac{\lambda^i}{i!}$ - For $n$ large and $\lambda$ moderate $$(1 - \frac{\lambda}{n})^n \approx e^{-\lambda}$$ $$\frac{n(n - 1)...(n - i + 1)}{n^i} \approx 1$$ $$(1 - \frac{\lambda}{n})^i \approx 1$$ Then $$P(X = i) \approx e^{-\lambda} \cdot \frac{\lambda^i}{i!}$$ - Expected Value & Variance $$E[X] = Var(X) = E[X^2] - (E[X])^2 = \lambda $$ $$E[X^2] = \lambda\cdot(1-\lambda)$$ - Computing the Poisson Distribution Function - If $X$ is Poisson with parameter $\lambda$, then $$\frac{P(X = i + 1)}{P(X = i)}= \frac{\lambda}{i+1}$$ - Starting with $P(X=0) = e^{-\lambda}$, we can use: $P(X = 1) = \lambda P(X = 0)$ $P(X = 2) = \frac{\lambda}{2} P(X = 1)$ $\vdots$ $P(X = i + 1) = \frac{\lambda}{i + 1} P(X = i)$ <!--ID: 1708098043672--> ## Cumulative Distribution Function - For a random variable X $F(x) = P(X \leq x), \quad -\infty < x < \infty$ is called the cumulative distribution function of X - 1. F is a non-decreasing function - 2. $\lim_{x \to \infty} F(x) = 1$ - 3. $\lim_{x \to -\infty} F(x) = 0$ - 4. F is right continuous. That is, for any $b$ and any decreasing sequence $b_n, n \geq 1,$ that converges to b, $\lim_{n \to \infty} F(b_n) = F(b)$ - $P(X \leq k) = e^{-\lambda} \cdot \sum_{i=0}^{k} \frac{\lambda^i}{i!}$ <!--ID: 1708098043677-->