Chapter 6: Random Variables and Probability Distributions Flashcards
Random Variable
takes numerical values that describe the outcomes of some chance process.
Probability Distribution of a Random Variable
gives its possible values and their probabilities
all probabilities
- 0<p<1
- add up to 1
Discrete Random Variable
doesn’t take a fixed set of possible values with gaps between
- has to be numeric
Probability Distribution Histogram
y-axis = probability
x-axis = values of x
Define Expected Value (mean) of a Discrete Random Variable
to find ______ of x, multiply each possible value by its probability, then add all the products.
mu of x = E(x) = XiPi + X2P2 + X3P3 … = Summation of XiPi
Interpretation of Expected Value (mean, mu)
If the random process of __context__ is repeated for a very large number of times, the average number of __x-context__ we can expect is __expected value__. (decimals OK)
ex. If the random process of __asking a student how many movies they watched this week__ is repeated for a very large number of times, the average number of __movies__ we can expect is __3.23 movies__.
Standard Deviation of a Discrete Random Variable
sigma (x) = sqrt (summation (xi - mu(x))^2Pi))
on the formula sheet I think
Interpretation of Standard Deviation of a Discrete Random Variable
The __context__ typically vary by __sigma__ from the mean of __mu__.
Variance of a Discrete Random Variable
sigma (x) ^2 = summation (xi - mu(x))^2Pi)
Continuous Random Variables
A random variable that can take any value in an interval
VARIABLE
What it describes: a characteristic of an individual
typical uses: analyzing the distribution of a specific sample
Types:
1. categorical
- possible values: any
- typical graphs used: pie chart, bar chart
2. quantitative
- possible values: numeric, where averages make sense
- typical graphs used: histogram, boxplot
examples:
variable: gender at birth
type: categorical
possible values: male, female
distribution: varies by sample
RANDOM VARIABLE
What it describes: a numerical outcome of a chance process
typical uses: answering questions of a chance process
Types:
1. discrete
- possible values: numeric with gaps between possible values
- typical graphs used: probability distribution table, histogram
2. continuous
- possible values: numeric on an interval of possible values
- typical graphs used: density curve
examples:
how many heads on 5 tosses of a coin?
Adding/Subtracting a Constant C, on a Probability Distribution
Shape: stays the same
Center: add/subtract C
Variability: stays the same
Multiplying/Dividing a Constant C, on a Probability Distribution
Shape: stays the same
Center: multiply/divide by C
Variability: multiply/divide by C
Adding/Subtracting Random Variables X and Y
Mean: mu (x±y) = mu (x) ± mu (y)
Standard deviation: sigma (x±y) = sqrt( sigma (x) ^2 + sigma (y) ^2) –> x and y must be independent random variables
If combine two normal (continuous) random variables
- any sum/difference of independent normal random variables is also normally distributed
- the mean and standard deviation of the resulting normal distribution can be found using the same rules for mean and variances that we have been using for discrete random variables.
Binomial Random Variable
Conditions: BINS
binomial formula
describing a binomial distribution: [shape, mean, standard deviation]
10% condition
large count condition
Binomial Conditions BINS
Binary: success and failures
Independent: each trial is independent of other trials
Number of trials: fixed
Same probability of success for each trial
Binomial Formula
P(X=x) = nCx * p^x * (1-p)^(n-x)
n = number of trials
x = number of success
p = probability of success
Describing a binomial distribution
shape: (when n is small)
- roughly symmetric when p is close to 0.5
- skewed right when p<0.5
- skwed left when p>0.5
mean: mu (x) = np
Standard deviation: sigma (x) = sqrt (np(1-p))
Binomial 10% and large count condition
10% condition: When sampling without replacement, we can treat individual trials as independent as long as n<1/10N
large count condition: np≥10 and n(1-p)≥10 –> can use normal distribution to model a dinomial distribution with mu = np and sigma = sqrt (np(1-p))
Geometric Random Variable
Conditions: BITS
geometric formula
describing a geometric distribution: [shape, mean, standard deviation]
Geometric Conditions BITS
Binary: success and failures
Independent: each trial is independent of other trials
Trials until first success
same probability of success for each trial
Geometric Formula
P(X=x) = (1-p)^(x-1)*p
p = probability of success
x = trial of first success
