# probability theory and statistics 8

- Prove that, if
*X*is Gaussian(*Î¼*, Ïƒ) and*Y*=*aX*+*b*, then*Y*is Gaussian(*aÎ¼*+*b*,*a*Ïƒ). - The peak temperature
*T*, in Fahrenheit, on a July day in Antarctica is a Gaussian random variable with Ïƒ2 = 225 and mean*Î¼*= âˆ’75. Calculate…- (a)
*P*(*T*> 0) - (b)
*P*(*T*< âˆ’100) - (c)
*P*(âˆ’90 â‰¤*T*< âˆ’60) - (d)
*P*(*T*> âˆ’75|*T*> âˆ’90) - (e) The probability that there is at least one day in July with temperature above 0 (assuming that the peak temperatures on different days are independent.)

- (a)
- Suppose
*X*is Gaussian(*Î¼*,Ïƒ).- (a) Find the probability of a positive â€œ4Ïƒ eventâ€, that is, observing
*X*>*Î¼*+ 4Ïƒ. Use the complementary CDF table posted on eLearning. - (b) Suppose you perform measurements on
*X*. If you perform 1000 measurements, what is the expected number of 4Ïƒ events? What is the probability of seeing at least one such event? - (c) In a series of independent measurements of
*X*, what is the expected number of trials required to accumulate a total of 10 4Ïƒ events? - (d) Derive the conditional probability that a â€œ4Ïƒ eventâ€, observing
*X*>*Î¼*+ 4Ïƒ, is actually a 4.5Ïƒ event,*X*>*Î¼*+ 4.5Ïƒ. Explain this results in terms of the shape of the â€œtailsâ€ of*f**X*.

- (a) Find the probability of a positive â€œ4Ïƒ eventâ€, that is, observing
- To win a prize at the fair, you need to hit the bullseye once with a bow and arrow. Your probability of success is only
*p*= 0.2 per shot. You get three shots; if you hit the bullseye, you stop shooting. Let*X*be the number of shots taken, and*Y*indicate winning the prize (*Y*= 1 for success,*Y*= 0 for failure.)- (a) Calculate the joint PMF of
*X*and*Y*. Report your result as a table, and confirm that your joint PMF is normalized. A tree diagram may be useful in working out the joint PMF. - (b) Calculate the marginal PMFs of
*X*and*Y*. - (c) Calculate Ï(
*X*,*Y*) and interpret its sign and magnitude. (Note: Ï(*X*,*Y*) and Ï*X*,*Y*are two different notations for the same thing, the correlation coefficient.)

- (a) Calculate the joint PMF of
- 40% of students own an iPhone, 20% of students own an Apple computer, and 16% of students own both. Let
*X*be a Bernoulli random variable that describes â€œowning an iPhone.â€, and let*Y*be a Bernoulli random variable that describes â€œowning an Apple computer.â€- (a) Determine the joint PMF of
*X*and*Y*. - (b) Calculate Ï(
*X*,*Y*) - (c) Calculate the conditional PMF
*P**X*|*B*(*x*), where*B*= {*Y*= 1}, the even that a student owns an Apple computer.

- (a) Determine the joint PMF of