Question

4. Suppose X has a discrete uniform distribution: the distribution function of X 5. A random variable Z has the pmf bclow. P (X-х,)-1 , is|2 n. Find 0 Pz(z) 0.20 0.16 0.4 a (1) What is thevalue of a ? (2) What is P(l S Z <3)? (3) What is Fz (1.7)? 6.

Answer 1

4) By definition of distribution function we have

$F_{X}(x_{i})=P(X\leq x_{i}) \\ ~~~~~~~~~~~~~ =P(X=x_{1})&plus;P(X=x_{2})&plus;\cdots &plus;P(X=x_{i}) \\ ~~~~~~~~~~~~~ ={1\over n}&plus;{1\over n}&plus;\cdots &plus;{1\over n} \\ ~~~~~~~~~~~~~ ={i\over n}$

5) a) Total probability is 1 so

$0.20&plus;0.16&plus;0.41&plus;a=1\\\Rightarrow a=0.23$

b)

$P(1\leq Z<3)=P(Z=1)&plus;P(Z=2)\\~~~~~~~ ~~~~~~~~~~~~~~~~~=0.16&plus;0.41\\~~~~~~~ ~~~~~~~~~~~~~~~~~=0.57$

c)

$F_{Z}(1.7)=P(Z=0)&plus;P(Z=1)\\~~~~~~~ ~~~~~~~~=0.20&plus;0.16\\~~~~~~~ ~~~~~~~~=0.36$