Discrete random variables Flashcards

1
Q

Define a discrete random variable

A

A random variable that can only take certain values in a finite or countable set

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2
Q

Define the probability mass function of a DRV

A

p(x) = P(X=x)

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3
Q

State two properties of pmfs

A

1) all p(x) ≥ 0
2) Sum of all p(x) = 1

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4
Q

Define the kth moment of X

A

E(Xk)

Similarly, the kth moment of X about a is E((X-a)k) (where a is a real number)

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5
Q

Give the expectation of E(XY) if X and Y are independent

A

E(XY) = E(X)E(Y)

Note: this property does not prove independence

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6
Q

Define the case for when rvs X and Y are independent

A

if P(X=x,Y=y) = P(X=x)P(Y=y) for all x and y, X and Y are independent

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7
Q

Give and describe Chebyshev’s inequality

A
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8
Q

Give and describe the weak law of large numbers

A

This law demonstrates that the long-run average of a random variable is very unlikely to be far from its expectation, as the variance of the long-run average gets smaller and smaller.

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9
Q

State the expectation of the Bernoulli distribution

A

p, where p is the probability of success

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10
Q

State the variance of the Bernoulli distribution

A

p(1-p), where p is the probability of success

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11
Q

Describe the Bernoulli random variable X and give the proper notation

A

X is defined as the number of successes we have if we perform the experiment.

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12
Q

Describe the Binomial random variable X and give its proper notation

A

X is defined as the number of successes we have counted after n independent Bernoulli trials have taken place

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13
Q

Give the probability mass function of the Binomial distribution

A

Note that the cdf for the binomial distribution is the sum of the individual probabilities.

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14
Q

State the E(X) for the binomial distribution

A

np

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15
Q

State the Var(X) of the binomial distribution

A

np(1-p)

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16
Q

Describe the geometric random variable X and give its proper notation

A

X is defined as the number of independent Bernoulli trials with the same probability of success up to and including the first success. X* may be used to denote “the number of failures until the first success”, and X* = X - 1

17
Q

Give the pmf for the geometric distribution

A
18
Q

State the E(X) for the geometric distribution

A
19
Q

State the Var(X) of the geometric distribution

A
20
Q

Describe the Poisson random variable X and give the proper notation

A

X is defined as the number of events we had recorded on one such surface we had available

21
Q

Give the pmf for the poisson distribution

A
22
Q

State the E(X) of the poisson distribution

A
23
Q

State the Var(X) of the Poisson distribution

A
24
Q

Describe how the Poisson distribution can be used to approximate the binomial distribution

A
  • This applies if n is large, p is small and np is moderate
  • μ = np
25
Q

Describe the negative binomial random variable X and give the proper notation

A

X is defined as the number of independent Bernoulli trials with the same probability of success, until we have r successes.

This is equivalent to r.v. X* for the number of faliures until the rth success, where X* = X - r

26
Q

Give the pmf of the negative binomial distribution

A
27
Q

State the E(X) of the negative binomial distribution

A
28
Q

State the Var(X) of the negative binomial distribution

A
29
Q

State the E(X) of the hypergeometric distribution

A
30
Q

State the Var(X) of the hypergeometric distribution

A

This does not need to be memorised

31
Q

Describe the hypergeometric random variable X and give its proper notation

A

X is defined as the number of units in the sample (of size n and that we selected without replacement) that have the characteristic

32
Q

Give the pmf of the hypergeometric distribution

A

Where we have N units, from which M share the same characteristic. We select n units from the population of N without replacement

33
Q

Describe how the hypergeometric and binomial distributions can be approximated.

A

When the ration of n/N is small, the hypergeometric (n,M,N) distribution may be approximated by the binomial (n,p) where p = M/N