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1. Let events An increase...

1. Let events An increase to A. Find A and show P(An ) -> P(A). Show the same thing in the decreasing case. Use these results to show that a df F is right continuous, F(oo)=1, F(-oo)=0.

2. Let X be exponential(2). Calculate the df, mgf and variance. Sketch the df.

3. Let X have pdf f(x)=c/x^2 , x>1 and is 0 otherwise. Find c. Calculate the first 2 moments.

4. Let X be standard exponential. Show P(X>s+t)=P(X>t|X>s), for all s,t>0. This is the ageless or memoryless property. Show this also holds for the geometric(p) if we require s,t to be positive integers.

5. Let U be a uniform((0,1)) rv. Set Y = !log(U). Calculate the pdf

of Y . Now suppose Z is Bernoulli(1/3). Find and sketch a function

g : (0,1) ! R such that Z d

= g(U). Hint: consider the function

g(u) = glb{z : F(z) # XXXXX}, 0 <u< 1. Here F is the df of Z.

6. .Two people each roll a fair 6 sided die. Let X be the number thrown

by the ﬁrst person and Y the number thrown by the second. You may

assume these are independent. Let V = |X!Y | and M = max{X, Y,}.

Calculate E(V ) and E(M).

7. Let X # XXXXX be either a counting or a continuous rv with df F. Assuming

E(X^2) = 2N) <= 1/4.

2. Let X be exponential(2). Calculate the df, mgf and variance. Sketch the df.

3. Let X have pdf f(x)=c/x^2 , x>1 and is 0 otherwise. Find c. Calculate the first 2 moments.

4. Let X be standard exponential. Show P(X>s+t)=P(X>t|X>s), for all s,t>0. This is the ageless or memoryless property. Show this also holds for the geometric(p) if we require s,t to be positive integers.

5. Let U be a uniform((0,1)) rv. Set Y = !log(U). Calculate the pdf

of Y . Now suppose Z is Bernoulli(1/3). Find and sketch a function

g : (0,1) ! R such that Z d

= g(U). Hint: consider the function

g(u) = glb{z : F(z) # XXXXX}, 0 <u< 1. Here F is the df of Z.

6. .Two people each roll a fair 6 sided die. Let X be the number thrown

by the ﬁrst person and Y the number thrown by the second. You may

assume these are independent. Let V = |X!Y | and M = max{X, Y,}.

Calculate E(V ) and E(M).

7. Let X # XXXXX be either a counting or a continuous rv with df F. Assuming

E(X^2) = 2N) <= 1/4.

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