Chapter 7: Random Variables and Discrete Probability Distributions Flashcards

1
Q

Random variable

A

A function or a rule that assigns a number to each outcome of an experiment

Name of random variable = uppercase letter

Value of random variable = lowercase letter

Probability X has the value of x = P(X= x) or just P(x)

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2
Q

Discrete random variable

A

Random variable that can take on a countable number of values

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3
Q

Continuous random variable

A

Random variable whose values are uncountable (could be anything - infinitely divisible)

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4
Q

Probability distribution

A

A table, formula, or graph that describes the values of a random variable and the probability associated with these values

When multiple outcomes are possible to reach a single value then the sum of these possible outcomes is the probability of that value

Probability distribution of a sample represents the probability distribution of the population

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5
Q

Requirements for a distribution of a discrete random variable

A

1) probability of x is between 0 and 1 for all values of x

2) some of the probability of all possible values of x = 1

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6
Q

Population mean

A

Aka expected value (of the random variable) E(X)

Weighted average of all values

Weight = probability

E(X) = sum of all individual values of x multiplied by their probability

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7
Q

Population variance

A

The weighted average of all the squared deviations from the mean

So sum of all difference between value of x and mean squared times the probability of that value of x

Shortcut:
Sum of all (values of x squared multiplied by the probability of x) less the mean squared

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8
Q

Laws of expected value

A

Where X is the random variable and c is a constant

E(c) = c
E(X+c) = E(X)+c
E(cX) = cE(X)
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9
Q

Laws of variance

A

Where X is the random variable and c is a constant

V(c)=0
V(X+c)= V(X)
V(cX) =c^2V(X)

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10
Q

Binomial experiment

A
  • consists of a fixed number of trials (number of trials = n)
  • each trial has two possible outcomes, one labeled success other labeled failure
  • probability of success is p. Probability of failure is 1-p
  • trials are independent (outcome of one trial does not affect the outcome of others)
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11
Q

Bernoulli Process

A
  • each trial has two possible outcomes, one labeled success other labeled failure
  • probability of success is p. Probability of failure is 1-p
  • trials are independent (outcome of one trial does not affect the outcome of others)

Binomial experiment without fixed number of trials

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12
Q

Binomial random variable

A

Random variable is the number of successes in the n trials

A discrete variable (only integers)

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13
Q

Binomial probability distribution

A

The probability of x successes in a binomial experiment with n trials (where the probability of success = p)

P(x)= (n!/ x!(n-x)!)p^x(1-p)^(n-x)

Remember 0!=1

(First half determines number of ways to get that number of successes second half determines probability of each way)

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14
Q

Cumulative probability

A

The probability that a random variable is less than or equal to a value

For value x finding P(X<=x)

Sum of probabilities of all random variable values x and below

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15
Q

Binomial table

A

Provides cumulative binomial probabilities for given values of n (tests) and p (probability of success in a single experiment)

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16
Q

Using binomial table to find the binomial probability of P(X>=x)

A

Basically using the compliment rule. The probability that X is greater than or equal to x value is 1 minus the probability it is less than or equal to x-1 so:

P(X>=x) = 1 - P(X<=[x-1])

17
Q

Using the binomial table to find the binomial probability or P(X=x)

A

P(X=x) = P(X<=x) - P(X<=[x-1])

18
Q

Binomial probability in excel

A

=BINOMDIST([x], [n], [p], True or False)

x = number of successes
n = total number of tests 
p = probability of success
True = cumulative probability (P(X<=x)
False = probability of individual value x
19
Q

Mean of a binomial random variable

A

= n * p

Number of tests * probability of success

20
Q

Variance of a binomial random variable

A

= np(1-p)

Number of tests * probability of success * probability of failure

21
Q

Standard distribution of a binomial random variable

A

Square root of variance

=√np(1-p)

22
Q

Poisson random variable

A

The number of successes that occur in a period of time or an interval of space in a Poisson experiment

23
Q

Poisson experiment

A

Characteristics:

  • number of success that occur in any interval is independent of the number of successes that occur in any other interval
  • the probability of a success in an interval is the same for all equal sized intervals
  • the probability of a success in an interval is proportional to the size of the interval
  • the probability of more than one success in an interval approaches 0 as the interval becomes smaller
24
Q

Poisson probability distribution

A

A discrete probability distribution

probability of x successes in a specific interval = e to the negative power of the mean number of successes times the mean number of successes to the power of x all divided by x factorial

P(x) = ((e^- mean)*(mean^x))/x!

e= base of the natural logarithm

25
Variance of a Poisson random variable
Equal to the mean of the random variable
26
Calculating the Poisson probability distribution when the interval does not match
- find the proportion of the interval of interest to the given mean interval - multiply or divide the mean by the relative proportion - solve qsbusyal
27
Cumulative Poisson probabilities by formula
Find each of the discrete probability and sun
28
Poisson table
Shows cumulative Poisson probabilities for selected mean values Gives values for P(x)<=x
29
Using Poisson table to find Poisson probability P(X>=x)
P(X>=x)= 1 - P(X<=[x-1])
30
Using Poisson probability table to find an individual value of x
P(X=x) = P(X<=x) - P(X<=[x-1])
31
Poisson probability in excel
=POISSON([x],[mean], True or False | True = cumulative probability, false = individual