Stats Week 7 Distributions PDF

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Professor Wang
Topic: Distributions, Binomial Distribution...

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Week 7 | Probability Distribution Sunday, May 23, 2021

2:51 PM

RANDOM VARIABLES -

Random variables- a variable whose value is determined by a chance event ○ Discrete random variable- numerical values that can be listed or counted § Ex. Number of phone calls made during the day, number of burgers sold ○ Continuous random variable- any values in an interval, often measurements § Ex. Exact length of time of a phone call, weight of a burger sold

DISTRIBUTIONS -

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Describe random outcomes, the possible outcomes, and the probability that each outcome will occur ○ Can be displayed in tables, graphs, or equations ○ All P values must be between 0 and 1 ○ Sum of all P values = 1 Probability distributions: ○ Outcomes described are random ○ A probability can be assigned to all possible outcomes ○ Probability of each possible outcome is the relative frequency of the outcome in a large number of repetitions ○ Sum of the probabilities of all possible outcomes =1 Discrete probability distribution ○ Ex. The number of times a student changes major at a college ○ P(x) where x is the number of times ○ Right skew ○ Best shown by a histogram Continuous probability distribution ○ Best shown by a curve with area under the curve equal to 1 ○ Probability of any single value occurring is zero

NORMAL DISTRIBUTIONS -

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Aka. Gaussian distribution Unimodal, symmetric, bell shaped curve ○ Center is the mean (mu, or u), defined by standard deviations (sigma) ○ Shape determined by the mean and sd (can have low or high spread/amplitude) 68% of data is within 1 SD 95% of the data is within 2 SD

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99.7% of the data is within 3 SD Probabilities can be found using a Z table and calculation of the z-score Percentiles ○ The nth percentile is a value where n% of the distribution is to the left of that point

BINOMIAL DISTRIBUTIONS

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Used for a random process with two possible outcomes (success and failure) ○ The binomial variable X is the number of successes in n repeated trials Used for discrete random variables only ○ Probability of success is the same in each trial ○ Trials are independent ○ Fixed number of trials Notation b(n,p,x)

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Expected value is the mean Standard deviation is how far values typically are from the mean

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Something unusual to observe would be 2+ standard deviations away from the mean The shape of the binomial distribution can be skewed or symmetric ○ All binomial distributions approach a normal distribution shape the large n is (more trials)...