12 Discrete Random Variables

Question 1

Random variable

Answer 1

A variable that takes on numerical values depending on the outcome of a random experiment.

Question 2

Discrete random variable

Answer 2

A random variable that can take no more than a countable number of values

Question 3

What is a random variable denoted by?

Answer 3

X

Question 4

What is P(X=x) or P(x)

Answer 4

The probability that X takes the specific value x

Question 5

Cumulative distribution function F(x)

Answer 5

Shows the probability that X is less than or equal to x

Question 6

How can the cumulative distribution function be written as a normal probability function

Answer 6

F(x)= P(X<=x)

Question 7

Joint probability function

Answer 7

Used to express the probability that X takes the specific value x and simultaneously Y takes the specific value y, as a function of x and y

Question 8

How is a joint probability function for x and y written

Answer 8

P(x,y) P(X=x n Y=y)

Question 9

Margin probabilities

Answer 9

Is the probability of one event happening irrespective of another event happening

Question 10

Conditional probability function

Answer 10

Expresses the probability that X takes the value x when the value y is specified for Y

Question 11

Equation for conditional probability function

Answer 11

P(x|y) =P(x,y)/P(y)

Question 12

What is the expected value

Answer 12

The analogous measure of central location for a random variable

Question 13

Properties of expected values

Answer 13

If X and Y are random variables and b is a constant
1. E(X+Y)= E(X)+E(Y)
2. E(bX)= bE(X)
3. E(b)= b
In general E(g(x)) and g(E(x)) are not equal

Question 14

Properties of the variance

Answer 14

If V, W and Z are random variables and b is a constant 
1. If Y=V+W:
Var(Y)= var(V)+var(W)+2cov(V,W)
2.if Y=bZ:
Var(Y)=b^2var(Z)
3. If Y=b:
Var(Y)=0

Question 15

What is the covariance when two random variables are statistically independent?

Answer 15

0

Question 16

Properties of covariance

Answer 16

1. If Y= V+W
Cov(X,Y)= cov(X,V)+cov(X,W)
2. If Y=bZ
Cov(X,Y)= bcov(X,Z)
3. If Y=b
Cov(X,Y)=0

Question 17

If the covariance is zero what does this tell us?

Answer 17

Nothing. The variables maybe independent or dependent

Question 18

Permutations

Answer 18

The number of possible oderings with a set of n objects and x ordered boxes

Question 19

Combinations

Answer 19

We are concerned with the number of different ways that x objects can be selected from n but not concerned about the order

Question 20

Which is bigger, the permutation or the combination?

Answer 20

Permutation

Question 21

Bernoulli distribution

Answer 21

A random experiment with only two possible outcomes of

Question 22

Binomial random variable

Answer 22

The outcome of a series of n independent Bernoulli trials

Question 23

Poisson distribution

Answer 23

The distribution of the number of times a certain event occurs in a specific time interval or in a specific length or area

Question 24

Assumptions of Poisson distribution

Answer 24

Assume the interval can be divided into very small sub intervals such that:

• the probability that an event occurs in one sub interval is very small
• the probability of one success in a sub interval is constant for all sub intervals and is proportional to its length
• the sub intervals are independent of each other

Question 25

What does lambda represent in Poisson distribution?

Answer 25

The mean number for successes in the subinterval and the variance

Question 26

Differences between Poisson and binomial

Answer 26

- a binomial is limited to the number of trials - a Poisson can take an infinite number of values - in binomial mean is greater than variance - in Poisson mean is equal to variance

Question 27

When can Poisson distribution be approximated to binomial distribution?

Answer 27

When n is large and p is small