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MTH4120: Midterm 1 Practice 2

Problem 1.

A fair die is rolled, and if the obtained number is divisible by \(3\), then the fair coin is tossed twice; if the obtained number is not divisible by \(3\), then the fair coin is tossed once. Let \(X\) be the random variable that represents the total number of heads obtained during this experiment.

  • (a) What is the sample space and the probability function that correspond to the experiment described in this problem?
  • (b) What is the expectation and the variance of \(X\)?

Problem 2. Assume that \(X\) is the number of heads obtained in \(n=5\) rolls of an unfair coin that shows heads with probability \(p=\frac34\). (In other words, \(X\) is a binomial random variable with parameters \(n=5\) and \(p=\frac34\).) Calculate the probability of the event \(\{X\leq 3\}\).

Problem 3.
  • (a) What is the definition of cumulative distribution function?
  • (b) A fair die is rolled once. If the obtained number is \(1\) or \(2\), then the player wins \(2\) dollars. If the obtained number is \(3\) or \(4\), then the player wins \(3\) dollars. If the obtained number is \(5\) then the player wins \(5\) dollars. If the obtained number is \(6\) then the player tosses a fair coin. If the coin lands heads, the player wins \(10\) dollars. However, if it lands tails, the player gets no money.
    Determine the cumulative distribution function for the random variable that represents the money that the player makes in this game. Draw the graph of the obtained function.

Problem 4. It is known that \(A\) and \(B\) are independent events that satisfy \(\mathbb P (A)=\frac{7}{25}\) and \(\mathbb P\left(A^C\cap B\right)=\frac15\). Calculate \(\mathbb P(B)\).

Problem 5. Assume that \(X\) is a random variable whose moment generating function satisfies the following: For every \(t\in (-\infty, 10)\) \[m_X(t)=\frac{100}{\left(10-t\right)^2}.\] Assume that \(Y\) is a random variable independent from \(X\) such that for every real number \(t\) the following holds: \(\mathbb P\left(X\leq t\right)=\mathbb P\left(Y\leq t\right)\). Evaluate \(\mathbb E\left[\left(2X+3Y\right)\cdot\left(3X-7Y\right)\right]\).

Problem 6. Eight boys and seven girls went to movies and sat in the same row of \(15\) seats. Assuming that all the \(15!\) permutations of their seating arrangements are equally probable, compute the expected number of pairs of neighbors of different genders.