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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Define the probability mass function of a DRV

A

p(x) = P(X=x)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

State two properties of pmfs

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

Give and describe Chebyshev’s inequality

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

State the expectation of the Bernoulli distribution

A

p, where p is the probability of success

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

State the variance of the Bernoulli distribution

A

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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

State the E(X) for the binomial distribution

A

np

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

State the Var(X) of the binomial distribution

A

np(1-p)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
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

18
Q

State the E(X) for the geometric distribution

19
Q

State the Var(X) of the geometric distribution

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

22
Q

State the E(X) of the poisson distribution

23
Q

State the Var(X) of the Poisson distribution

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
Describe the negative binomial random variable X and give the proper notation
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
Give the pmf of the negative binomial distribution
27
State the E(X) of the negative binomial distribution
28
State the Var(X) of the negative binomial distribution
29
State the E(X) of the hypergeometric distribution
30
State the Var(X) of the hypergeometric distribution
This does not need to be memorised
31
Describe the hypergeometric random variable X and give its proper notation
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
Give the pmf of the hypergeometric distribution
Where we have N units, from which M share the same characteristic. We select n units from the population of N without replacement
33
Describe how the hypergeometric and binomial distributions can be approximated.
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