# Probability of Outcomes of Different Situations

Five percent of patients suffering from a certain disease are selected to undergo a new treatment that is believed to increase the recovery rate from 30% to 50%. A person is randomly selected from these patients after the completion of the treatment and is found to have recovered. What is the probability that the patient received the new treatment?

An executive has three telephones on her desk, each with a different number, and none is an extension. During a 15-second interval the probability that phone 1 rings is .35, that phone 2 rings is .60, and that phone 3 rings is .10. find the probability that during this interval of time

All three phones ring.

At least one phone rings.

At most two phones ring.

None of the phones ring.

Let ƒ be a symmetric density. Let a > 0. Show that:

F(0)=1/2, where F is the distribution function.

P(X>a)=1/2- where a=0, b=a.

F(-a) + F(a) =1.

P(lXl > a) = 2F(-a) = 2[1-F(a)].

P(lXl < a) = 2F(a) -1.

Which of the following functions are joint density functions?

f(x,y)=e-(x+y), x>0, y>0, and zero elsewhere.

f(x,y)=8x2y, 0<y<x<1, and zero elsewhere.

A point X is chosen from the interval [0,1] according to the density function ƒ1(given below). Another point Y is chosen from the interval [X,1] (so that X< Y< 1) according to the density function g(ylx). Let

ƒ1(x) =1, 0<x<1, and zero elsewhere, and for 0< x < 1

g(ylx)= 1/(1-x) if x< y <1, and zero elsewhere.

Find the joint density of X and Y and the marginal density of Y.

Find the conditional density of X given Y=y.

A fair die is rolled twice. Let X be the sum of face values and Y the absolute value of the difference in face values on the two rolls. Are X and Y independent? Please justify the answer.

The acidity of a compound X depends on the proportion Y of a certain chemical present in the compound. Suppose

X = (α + βY)2

Where α, β are constants. If the density of Y can be assumed to be

g( y) = 2y, 0< y < 1, and zero elsewhere

Find the average acidity of the compound and Var(X).

For each of the following distributions, find a (the) median.

f (x) = 1/ (σ√2π)exp{-(x- μ)2/ 2 σ2}; -∞ < x <∞, μ ∈R

f(x)= 1/x2, x > 1, and zero elsewhere.

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The solution discusses the probability of outcomes of different situations.