Probability

Question 1

What is the equation for normalization?

Answer 1

sum ove i of Pi = 1

Question 2

What is the equation for the mean or average or expected value?

Answer 2

= sum over i of xi*Pi

Question 3

What is the equation for the variance?

Answer 3

σ(x)^2 = )^2> = -^2

Question 4

What is the equation for the standard or RMS deviation?

Answer 4

σ(x) = sqrt(variance)

Question 5

What are the variables in discrete probability distribution?

Answer 5

x is a discrete random variable with values xi and probabilities Pi

Question 6

What are the variables in continuous probability distributions?

Answer 6

x is a continuous random variable with probability P(x)dx of having a value between x and x+dx

Question 7

What is the equation for normalization in continuous probability distribution?

Answer 7

integral of P(x) dx = 1

Question 8

What is the equation for the mean in continuous probability distribution?

Answer 8

= integral of x*P(x) dx

Question 9

What is a Bernoulli trial?

Answer 9

An experiment with two outcomes.

Question 10

What are the two probabilities for a Bernoulli trial?

Answer 10

Pi(x) = P for xi = 1 and 1-P for xi = 0

Question 11

What is a good example of a Bernoulli trial?

Answer 11

Tossing a coin.

Question 12

What is the equation for the mean value of x for tossing a coin?

Answer 12

= sum of xiPi = 1P + 0*(1-P) = P

Question 13

What is the equation for the mean value of the squared x values for the coin tossing example?

Answer 13

= sum of xi^2Pi = 1P+0*(1-P) = P

Question 14

What is the equation for the standard deviation for the coin toss example?

Answer 14

σ(x) = sqrt(-^2) = sqrt(P-P^2) = sqrt(P*(1-P))

Question 15

If we perform the same experiment n times, what is the probability of getting k successes and n-k failures?

Answer 15

Given by the binomial distribution: P(n,k) = nCkP^k(1-P)^(n-k), where nCk = n!/(k!*(n-k)!)

Question 16

In Bayes theorem, what is the condition probability?

Answer 16

P(A|B) = prob of event A given that event B has occurred.

Question 17

In Bayes theorem, what is the joint probability?

Answer 17

P(A n B) = prob that events A and B occur = P(A)P(B) if A and B are independent -> if they are not independent: = P(B|A)P(A)

Question 18

What is the equated version of the joint probability?

Answer 18

P(A|B) = P(B|A)*P(A)/P(B) = Bayes theorem

Question 19

An athlete fails a drugs test which is 95% accurate. What is the likelihood they are guilty if on average only 1% of athletes take drugs? (first step)

Answer 19

P(D) = 0.01, P(\D) = 0.99, so P(+|D) = 0.95, P(+|\D) = 0.05, P(-|\D) = 0.95, P(-|D) = 0.05

Question 20

An athlete fails a drugs test which is 95% accurate. What is the likelihood they are guilty if on average only 1% of athletes take drugs? (second step)

Answer 20

Find P(+) = P(+|D)P(D)+P(+|\D)P(\D) = 0.06

Question 21

An athlete fails a drugs test which is 95% accurate. What is the likelihood they are guilty if on average only 1% of athletes take drugs? (third step)

Answer 21

Bayes theorem: P(D|+) =P(+|D)P(D)/P(+) = 0.950.01/0.06 = 0.16, so ~ 16%