Continuous Random Variables Flashcards

1
Q

Define a continuous rv

A

A random variables that has an infinite amount of outcomes. Note that probability is assigned to a range, the probability of any given value = 0

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2
Q

Define a probability density function

A

A pdf is a function f that satisfies:

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3
Q

Define the cumulative distribution function F

A

note that we can find f(x) from F(x) by differenting

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4
Q

Give four properties of the cdf

A
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5
Q

Give E(X) if X is a crv

A

Note: the integral must be < ∞

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6
Q

State the E(Y) if Y = g(X)

A
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7
Q

Give Var(X) if X is a crv

A

Note that the standard deviation is the positive square root of the variance

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8
Q

Give four properties of expectation and variance

A
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9
Q

Define the median of a crv

A
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10
Q

Define the mode of a crv

A

This can be found by finding the stationary points of the pdf

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11
Q

Define the case for which two crvs, X and Y, are independent

A

Note that E(XY) = E(X)E(Y) still holds for crvs

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12
Q

Give the formulae for the moment generating function, M(t)

A
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13
Q

Describe how to obtain E(Xk) using the mgf

A
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14
Q

Give the mgf of Y where Y = aX + b, given that X has mgf Mx(t)

A
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15
Q

Give the proper notation of the uniform distribution over the interval a,b

A
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16
Q

Give the pdf of a uniformly distributed crv over the interval a,b

17
Q

State the E(X) of the uniform distribution over the interval a,b

18
Q

Describe the exponential random variable and give its proper notation

A

X is defined as the continuous time or interval in space until the first event takes place

19
Q

Give the CDF of the exponential distribution

20
Q

Give the pdf of the exponential distribution

21
Q

State the E(X) of the exponential distribution

22
Q

State the Var(X) of the exponential distribution

23
Q

Give the pdf of the gaussian distribution

A

Note: the Gaussian distribution is the same as the normal distribution

24
Q

Give the proper notation for the normal/gaussian distribution

25
State E(X) for the gaussian distribution
26
State Var(X) for the Gaussian distribution
27
State the parameters of the standard normal distribution
µ = 0 σ2 = 1
28
Give the pdf for the standard normal distribution
29
Give the cdf of the standard normal distribution
30
Give Z such that X~N(µ,σ2) and Z~N(0,1)
This is the equivalent to Z = (X-µ)/σ/√n
31
Give the equivalent of Φ(-z) due to symmetry
1-Φ(z)
32
Describe the central limit theorem
The sum of a large number of independent and identically distributed random variables from any distribution is approximately normally distributed.
33
Describe the gamma random variable and give its proper notation
When there is a rate λ \> 0 of rare events per unit of time or space, X is defined to be the continuous interval of time or space until the α-th event takes place, where α is a positive integer. Note that the gamma distribution is the continuous analogue of a Negative Binomial rv α is known as the index or shape parameter λ is known as the scale parameter, or the 'parameter'
34
Define the gamma function Γ(.)
35
Give the pdf of the gamma distribution
36
State E(X) for the gamma distribution
37
State the Var(X) of the gamma distribution
38
State the Var(X) of the uniform distribution