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In: Statistics and Probability

1. Determine each of the following is true or false? If false, provide a counterexample. (a)...

1. Determine each of the following is true or false? If false, provide a counterexample.
(a) Let X be a continuous random variable which has the pdf fX. Then, for each x, 0 ≤ fX(x) ≤ 1.

(b) Any two independent random variables have ρXY = 0.

(c) Let X and Y be random variables such that E[XY ] = E[X]E[Y ]. Then, X and Y are independent.

2. Ann plays a game with Bob. Ann draws a number X1 ∼ U(0,1) and Bob draws a number X2 ∼ U(0,1). Assume X1 and X2 are independent.
(a) Calculate the conditional probability of Ann winning the game given Ann draws x1 ∈ [0,1].

(b) Calculate the probability of Ann winning the game. Hint: this is equal to P(X1 > X2). You may calculate this directly using integration; An alternative way is to use a geometric intuition. If x-axis represents x1 and y-axis represents x2, what does the set of (x1,x2) such that x1 > x2 look like? What is the size of this set?

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orchestra answered 1 year ago
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