Ch 3

Question 1

When is a continuous rv uniform(X~Unif(a,b)?

Answer 1

If X has a constant pdf
fX(x)=1/(b-a), for a<=x<=b
0, otherwise

Question 2

How do we define the indicator function?

Answer 2

I(S) :=(1, S is true,
0, S is false.
Sometimes the statement S is implicit and we write for a set A
IA (x) := I(x ∈ A) =(1, x ∈ A,
0, x ∉ A

Question 3

Define a Bernoulli random variable. (X~Bern(p))

Answer 3

A Bernoulli random variable with parameter p (where 0 ≤ p ≤ 1) is a random variable such that P (X = 1) = p
and P (X = 0) = 1 − p

Question 4

Explain simply what a Bernoulli random variable is.

Answer 4

A one off trial that succeeds with probability p

Question 5

Define Binomial distribution. (X~Binom(n,p))

Answer 5

The probability of getting exactly
k successes in n trials is given by the binomial probability mass function (PMF):P(X=k)=nCkp^k(1-p)^(n-k)

Where:•the binomial coefficient, which gives the number of ways to choose
k successes from n trials
•p is the probability of success on any given trial.
•(1−p) is the probability of failure.
•k is the number of successes (where
k=0,1,2,…,n).

Question 6

Explain simply the binomial distribution.

Answer 6

a discrete probability distribution that describes the number of successes in a fixed number of independent and identical trials of a binary experiment, where each trial has only two possible outcomes: success or failure. The binomial distribution is characterized by the number of trials n and the probability of success p in each trial

Question 7

Does the binomial distribution model a discrete or continuous random variable?

Answer 7

Discrete( number of successes in n trials)

Question 8

Define geometric distribution(X~Geom(p))

Answer 8

P(X=k)=(1-p)^(k-1)*p

Question 9

Explain simply what geometric distribution is.

Answer 9

a discrete probability distribution that models the number of trials needed to get the first success in a sequence of independent and identical Bernoulli trials

Question 10

Explain the memoryless quality of geometric distribution

Answer 10

For all k,j€{1,2,…} P(X=k+j|X>j)=P(X=k)
If we have already had j failures without a success then the probability of getting a success in k tries is the same as if we had just started.

Question 11

Define Poisson distribution(X~Pois(lambda))

Answer 11

A random variable X such that for k ∈ {0, 1, 2, . . . }
P (X = k) = e^(-λ)*λ^k/k!

Question 12

Explain Poisson distribution simply

Answer 12

a discrete probability distribution that models the number of events occurring within a fixed interval of time or space, under the assumption that these events occur independently and at a constant average rate

Question 13

What does lambda represent in poisson?

Answer 13

n*p

Question 14

Define an exponential random variable (X~exp(lambda))

Answer 14

A random variable X with pdf
fX (x) = λe^(−λx)1[0,∞)(x) =
(0, x < 0,
λe^(−λx) , x ≥ 0,
where λ > 0,

Question 15

Explain simply exponential distribution

Answer 15

a continuous random variable that models the time between events in a Poisson process, which describes a series of events happening independently at a constant average rate

Question 16

What is the cdf of the pdf of an exponential random variable?

Answer 16

Let X ∼ Exp(λ) for λ > 0. Then the cdf of X is given byFX (x) =1 − e^(−λx) 1[0,∞)(x).

Question 17

Explain the memoryless quality of exponential distribution

Answer 17

for all t , s ≥ 0
P (X > t + s|X > s) = P (X > t ) .
This means that when we have already waited for s time units and nothing happened then the probability of
something happening in the next t time units is the same as if we had just started. I.e. there is no memory of
already elapsed time.

Question 18

When are two random variables independent?

Answer 18

We call X and Y independent if for all sets A, B we have
P (X ∈ A and Y ∈ B) = P (X ∈ A) P (Y ∈ B) .

Question 19

If X an Y are discrete random variables when are they independent?

Answer 19

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