Question

PHYS10471 2. The following equation occurs in conditional probability: a) 5 marks] B marks 2 marks] b) An electronic system contains in components which are connected in series and they i) Describe the meaning of each term in the equation ii) Write down an expression for the unconditional probability P() in terms of iii) Describe the implications omrix)>MYIX). quantities in the above equation. function independently of each other. The length of time for each component until failure follows an exponential distribution density function with mean Determine the mean and variance of the length of time until failure of the system. [5 marks] c) Suppose that 10% of people in a certain population have the eye disease glaucoma. For persons who have glaucoma, measurements of eye pressure x will be normally distributed with a mean of 25 mm-Hg and a variance of 1 mm-Hg. For persons who do not have glaucoma, the pressure will be normally distributed with a mean of 20 mm-Hg and a variance of 1 mm-Hg. Suppose that a person is selected from the population and her eye pressure measured. Determine the conditional probability that the person has glaucoma given [5 marks] i) pressure x. ii) For what value of x is the conditional probability in i) greater than 1/2? 5 marks]

Answer 1

2 a)

(i)

P(X|Y) = conditional probability that event X occurs given that event Y occurs

P(Y|X) = conditional probability that event Y occurs given that event X occurs

P(X) = unconditional probability that event X occurs

P(Y|) = conditional probability that event Y occurs given that event X does not occur

P() = unconditional probability that event X does not occur

(ii)

P(Y) = P(YX) + P(Y)

P(Y) = P(Y|X)P(X) + P(Y|)P()

P(Y) = P(X)}

(iii)

P(Y|) = P(Y|X)

conditional probability that event Y occurs given that event X does not occur

= conditional probability that event Y occurs given that event X occurs

Now ; P(Y|) = P(Y|X)

P(X) P(X) + P(X) P(X|Y) = P(YX)PX) + P(Y|X)P(X)

i.e ; X and Y are independent events   

2 b)

2c)

(i)

(ii)