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\title{
Exam for Probability \& Statistics (6PAS1) \\ winter semester 2025/2026
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\section*{Questions set no. 2}

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\textbf{1.} 
Define covariance between two random variables. Write and prove an alternative 
formula for covariance which is easier for computations.


\bigskip


\textbf{2.}
Three players play a game of throwing a coin. Those who get more heads, wins. 
Each player throws up to three times, but the first player is a big pessimist: 
if at the first throw he get tail, he stops throwing. The other player is also a
pessimist, but a lesser one: he stops throwing if he get tail in the first two 
throws. The third player never stops prematurely: he always throws a coin three
times. Let $X$, $Y$ and $Z$ be random variables, equal to how many heads got 
the first, the second, and the third player, respectively. Determine:
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a) $E[X]$ 
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b) $E[Y]$ 
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c) $E[Z]$ 
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d) $P(X < Y)$ .



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