Random Variables

random variable

assigns a number to outcomes of a random experiment

discrete RV

takes values from a countable set

continuous RV

takes values in an interval of real numbers

discrete

Is this example discrete or continuous? Number of packet arrivals in 1 ms.

discrete

Is this example discrete or continuous? Number of bit errors in a transmitted frame.

continuous

Is this example discrete or continuous? Amplitude of thermal noise at a receiver.

continuous

Is this example discrete or continuous? Time between successive packet arrivals.

mean

the average or center of distribution (µ)

variance

measures how spread out the values are (σ^2)

sample space

the set of all possible outcomes (Ω)

one realization of a random experiment

Each outcome within a sample set corresponds to…

events

Probabilities are assigned to ______, which are subsets of a sample set.

probability mass function (pmf)

assigns a probability to each possible value

discrete

What type of RV is pmf used for?

P(X=x)

px(X) = ?

countable

pmf is used when there is a __________ set of outcomes of a random variable.

px(1)=0.5, px(0)=0.5

Let X represent the outcome of a fair coin: X=1 is heads, X=0 is tails. What is the pmf?

Bernoulli random variable

models a single trial with only two outcomes

p

For a Bernoulli RV, what is the probability of outcome=1 (a success)?

1-p

For a Bernoulli RV, what is the probability of outcome=0 (a fail)?

Bernoulli

What type of random variable is this example? One bit is being transmitted over a noisy channel. X=1 if a bit is received in error (probability p = 0.1) and X = 0 if a bit is received correctly (probability 0.9).

binomial random variable

counts the number of successes in n independent Bernoulli trials

binomial coefficient

counts how many ways to have exactly k successes

binomial RV

P(X = k) = (n k)*p^k*(1 − p)^n−k

binomial coefficient

(n k) = n! / k!(n − k)!

probability of k successes

What does p^k represent in binomial RV?

probability of failures

What does (1-p)^n-k represent in binomial RV?

pmf

The equation for binomial RV can be used to find the _____ in a discrete case.

probability density function (pdf)

describes the density of probability at a point for a continuous RV (where the RV is likely to occur)

P(a ≤ X ≤ b) = integral from a to b (fx(x)dx)

PDF

What function is used for both discrete and continuous RVs?

probability distribution function (PDF)

describes the cumulative probability up to a point x

discrete

What type of RV is this PDF used for Fx(x) = sum of px(t)?

continuous

What type of RV is this PDF used for Fx(x) = integral from −∞ to x of fx(t)dt?