Math 375 Exam 3 1. (10 points) Let X be a continuous random variable with the pr
ID: 3319993 • Letter: M
Question
Math 375 Exam 3 1. (10 points) Let X be a continuous random variable with the probability density function given by y(z) 4ra if 0 z 1 and 0 otherwise. (a) Find P(XS 1 I x-b. (b) Find the expectation and variance of x 2. (5 points) Let X be a normal random variable with mean 12 and variance 4 Find the value of e such that P(X>c)-0.1 normally distributed. 3. (10 points) If 25 percent of these physicians ean less than $180,000 and 25 percent earn more The salaries of physicians are approximatel than $320,000, approximately what fraction earn (a) less than $200,000 (b) between $280,000 and $320,000? 4. (10 points) Let X and Y be two continuous random variables with joint prob- ability density function fxr(x,y) ry ifosvsrsl and 0 otherwise (a) Find the constant c. (b) Find fx(x) and fy (v). (c) Find P(YS YS ). 5. (15 points) Consider the unit disk D f(,y): +' s1). Suppose we choose a point (r, y) uniformly at random in D. That is, the joint probability density function of X and Y is given by fry(z,v) = c if (z,v) e D and 0 otherwise. (a) Find the constant c. (b) Find fx(x) and fy(v). (c) Are X and Y independent? Explain your answer.Explanation / Answer
3) here mean salary from above given data =(180000+320000)/2 =250000
for first quartile ; z score =-0.6745
therefore 320000 =mean +z*std deviation
320000=250000+0.6745*std deviaiton
std deviaiton =70000/0.6745=103782.2
a) probability that salary is les than 200000=P(X<200000)=P(Z<200000-250000)/103782.2)=P(Z<-0.4818)=0.3150
b)
P(280000<X<320000)=P((280000-250000)/103782.2<Z<(320000-250000)/103782.2)
=P(0.2891<Z<0.6745)=0.75-0.6137 =0.1363