Topcis

Discrete and Continuous Random Variables
- A random variable takes numerical values determined by the outcome of a chance process. The probability distribution of a random variable tells us the possible values
  - A discrete random variable has a fixed set of possible values with gaps between them.
    - The mean is defined by $\mu_x=\sum{x_i \cdot p_i}$
    - The standard deviation is defined by $\sigma_x=\sqrt{\sum{(x_i-\mu_x)^2}p_i}$
  - A continuous random variable takes all values in an interval. A denstiy curve is used to describe the distribution.
- The mean of a random variable is the balance point of the probability distribution histogram or density curve. The mean is also known as the expected value
Transforming and Combining Random Variables
- (+) or (-) two random variables by b will increase/decrease the $\mu$ by b but doesn't change the $\sigma$
- (x) or (/) two random variables by b multiplies/divides the $\mu$ and $\sigma$ by b but doesn't change the shape of the probability distribution
- A linear transformation of a random variable can be written as $Y=a+bX$
  - Shape: remains the same if $b>0$
  - Center: $\mu_y=a+b\mu_x$
  - Spread: $\sigma_y=|b|\sigma_x$
- If X and Y are any two random variablesL
  - $\mu_{x \pm y}=\mu_x\pm \mu_y$
- If X and Y are independent random variables, variances add
  - $\sigma^2_{x \pm y}=\sigma^2_x+\sigma^2_y$
Binomial and Geometric Random Variables
- A binomial setting counts the number of successess (binomial random variable) for the binomial distribution
  - If the size of the population isn't constant, the sample size must be no more then 10% of the population
- A geometric setting finds the probability of the first success (geometric random variable) in a geometric distribution

Formulas

$\LARGE \mu_x=\sum{x_i \cdot p_i} \\[20pt] \LARGE \sigma_x=\sqrt{\sum{(x_i-\mu_x)^2}p_i} \\[20pt] \LARGE \binom{n}{k}=\frac{n!}{(n-k)!k!} \\[20pt]$

Binomial Distribution Formulas

$\LARGE P(X=k)=\binom{n}{k}p^k(1-p)^{n-k} \\[20pt] \LARGE \mu_x=np \\[20pt] \LARGE \sigma_x=\sqrt{np(1-p)} \\[20pt] \LARGE n \leq \frac{1}{10}N \\[20pt]$

Geometric Distribution Formulas

$\LARGE P(Y=k)=(1-p)^{k-1}p \\[20pt] \LARGE \mu_y=\frac{1}{p} \\[20pt]$

Terms

Review

You want to find the expected value of flipping a coin 20 times. Which formula should you use?
- A) $\mu_x=\sum{x_i \cdot p_i}$
- B) $\mu_x=np$
- C) $\mu_x=\frac{1}{p}$
- D) $\mu_x=\frac{\sum{s}}{n}$
You want to find the probability of success for the nth attempt. What type of distribution is this?
- A) Probability distribution
- B) Binomial distribution
- C) Geometric distribution
- D) Guassian distribution
Let X equal the number of rebels Darth Vader kills each month and Y equal the number of Stormtroopers Luke kills each month. If $T=X-Y$ , how do you find $\sigma_T$ ?
- A) $\sigma_t=\sigma_x + \sigma_y$
- B) $\sigma_t=\sigma_x - \sigma_y$
- C) $\sigma_t^2=\sigma_x^2 + \sigma_y^2$
- D) $\sigma_t^2=\sigma_x^2 - \sigma_y^2$
You won a carnival game on your 3rd attempt but are accused of cheating since it takes most people 20 tries. If the probability of winning is 1/20, what is the probability of winning on your 3rd attempt? (enter as a decimal rounded to the nearest hundreth)
Amazon's autonomous robots have a 2% chance of misidentifying a human for a box. If the robot scans 50 things, what is the expected number of humans misidentified?