Discrete Probablity Distribution

Question 1

What is a Discrete probability distirbution?

Answer 1

A table or formula listing all possible values that a discrete r.v. can take, together with the associated probabilities

Question 2

How is the central tendency of a discrete probability distribution described?

Answer 2

• Expected value or mean of a discrete random variable, X, which takes on values x with probability p(xi) is:
• µ = E(X) = [Σall xi] xi * p(xi)

Question 3

What are the rules for expectation?

Answer 3

• E(c) = c
• E(cX) = cE(X)
• E(X-Y) = E(X) - E(Y)
• E(X + Y) = E(X) + E(Y)
• E(XY) = E(X) * E(Y) only if X & Y are independent

Question 4

How is the variance of a discrete probability distribution described?

Answer 4

• σ^2 = E[ (X - µ)^2 ]
  
  • Expected value of the square distance from the mean
• σ^2 =Σall-xi [ x^2 * p(xi) ] - µ^2

Question 5

What are marginal distribution functions?

Answer 5

• For example, P(X=1) = p(1,0) + p(1,1) + p(1,2)
• In general, marginal distribution of X is given by
  
  • p(x) = P(X = x) = Σ p(x,y)
• Must be drawn as separate table

Question 6

How is independence of random variable’s ascertained?

Answer 6

• If independent, then the joint probability (probability of the intersection) will be equal to the product of the marginal probabilities (row value x column value = cell value then independent)
• If random variables X & Y are independent, then:
  • P(X=x and Y=x) = P(X = x) * P(Y=y)
  • p(x,y) = px(x) * py(y)

Question 7

What is the covariance of a discrete probably distribution?

Answer 7

• Measures how the random variable move together

- If move together in same direction: +, if opposite: -

Question 8

What is the correlation coefficient of a discrete probability distribution?

Answer 8

• Correlation coefficient
  
  • Associated with covariance
  • ? = (cov (X,Y)) / σx σy
  • 1 ≤ ? ≤1
  • If X and Y are uncorrelated, does not mean independence
  • If X and Y are independent, they will always have 0 correlation

Question 9

What are the properties of a binomial experiment?

Answer 9

• A fixed number of trials, n
• Two possible outcomes for each trial, labelled success and failure
• Probability of success is p, probability of failure is (1-p)
• Trials are independent - any outcome of one trial does not affect the outcomes of any other trials

Question 10

What is a binomial random variable?

Answer 10

The binomial random variable is defined as the number of “successes” in the n trials

Question 11

What does nCk mean?

Answer 11

the number of ways to arrange k things out of n things, without regard to order (when the k things are identical)

Question 12

How do we notate Combinations?

Answer 12

• If X is a binomial random variable with n trials and p is the
  probability of success in each trial, and we are interested in counting up the number of successes in those n trials, if X is number of successes then we write X~Bin(n,p)
• P(X=k) when X~Bin(n,p) means the probability that we will have k successes and (n-k) failures; we know how many ways we arrange those successes and failures

Question 13

What good is the binomial formula?

Answer 13

Given any number of trials, we can find the probability of observing k successes so long as we know the probability of success in each trial

Question 14

What are the mean and variance of a binomial distribution?

Answer 14

• If X~Bin(n,p), then it can be shown that
• µx = E(X) = np
  
  • Infinitely long run average
• σ2x = np(1-p)
  
  • Std.Dev square root
  • Biggest variance we can get is with p = 0.5

Question 15

What is important about the binomial tables?

Answer 15

Cannot interpolate! Only use tables is exact n and p are available