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MTH4120: Midterm 1 Practice 3

Problem 1.

It is known that \((\Omega,\mathcal F,\mathbb P)\) is a probability space. It is known that and \(M\) and \(N\) are two events on this probability space. Which of the following objects is a properly defined real number?

  • (A) \(M+\mathbb P(N)\)
  • (B) \(\mathbb P(M\cap N)\)
  • (C) \(\mathbb P\Big(\left\{M=10\right\}\cup\left\{N=11\right\}\Big)\)
  • (D) \(\mathbb P\Big( M \cup\left\{N=11\right\}\Big)\)
  • (E) \(\mathbb E[M]+\mathbb P(N)\)

Problem 2.

The sample space \(\Omega\) is given by \(\Omega=\Big\{ y, \{x\}, \{x,y\}, \{y,\{x,y\}\} \Big\}\). Which of the following objects is an outcome but not an event?

  • (A) \(\mathbb P:\Omega\to [0,1]\)
  • (B) \(\mathbb P:\mathcal F\to [0,1]\)
  • (C) \(\{x,y\}\)
  • (D) \(\{\{x\}, y\}\)
  • (E) \(\{y,\{x,y\}\}\)

Problem 3. The events \(A\) and \(B\) on probability space \((\Omega, \mathcal F,\mathbb P)\) satisfy \(\mathbb P\left(A^C\cap B\right)=0.31\), \(\mathbb P\left(A\cup B\right)=0.79\), and \(\mathbb P\left(A\cap B\right)=0.25\). Evaluate \(100\cdot \mathbb P\left(A\right)\).

Problem 4. A group of friends plans to visit only these four groups of animals at the zoo: aligators, bees, camels, and dolphins. They are deciding on the order in which the animals should be visited. One of the following schedules will be chosen at random:
  • 1. Visit the aligators first, then bees, then camels, and finally dolphins.
  • 2. Visit the dolphins, then aligators, then camels, then bees.
  • 3. Visit the camels, then dolphins, then aligators, then bees.
  • 4. Visit the bees, then aligators, then dolphins, then camels.
  • 5. Visit the camels, then bees, then aligators, then dolphins.
Each of the above five plans has equal chance of being selected. Construct the probability space \((\Omega,\mathcal F, \mathbb P)\) that corresponds to the described random experiment. Determine the set that corresponds to the event that the dolphins are visited before the aligators.

Problem 5. The random variables \(X\) and \(Y\) are independent. The codomain of both random variables \(X\) and \(Y\) is the set \(\{10,11,12,\dots, 110\}\). It is known that \(\mathbb P(X=10)= \frac{ 3 }{11}\), \(\mathbb P(Y=11)= \frac{ 2 }{11}\), and \(\mathbb P\left(\left.Y >11\right|X >12\right)=\frac{15}{22}\). What is \(242\cdot \mathbb P(X+Y=20)\)?

Problem 6. Evaluate \(18\cdot \mathbb P(X=0)+90\cdot \mathbb P\left(X\in(2,4]\right)-18\cdot \mathbb P\left(X\in(3,5]\right)\), where \(X\) is a random variable whose cumulative distribution function \(F_X(t)\) has the graph given in the picture below.

Problem 7. The probability density function \(f_X\) of the continuous random variable \(X\) satisfies \[f_X(t)=\left\{\begin{array}{ll} 0, &\mbox{if } t\leq -14;\\ \frac{t+14}{196},&\mbox{if }t\in\left(-14,0\right];\\ \frac{14-t}{196},&\mbox{if }t\in\left(0,14\right);\\ 0,&\mbox{if }t\geq 14. \end{array}\right.\] Evaluate \(196\cdot \mathbb P\left(X\in \left(-5,9\right)\right)\).

Problem 8. Prove that there exists a probability space \((\Omega, \mathcal F, \mathbb P)\) and two random variables \(X\) and \(Y\) on this probability space such that \[\mathbb E\left[X\right]=4,\quad \mathbb E\left[Y\right]=7,\quad \mbox{ and }\quad \mathbb E\left[XY\right]=2.\]