Next, run a computer simulation to carry out this experiment. One ticket will win \(\$1,000\), two tickets will win \(\$500\) each, and ten tickets will win \(\$100\) each. Find the probability that the next litter will produce five to seven live pups. \[\begin{array}{c|c c c c} x &-2 &0 &2 &4 \\ \hline P(x) &0.3 &0.5 &0.2 &0.1\\ \end{array}\], \[\begin{array}{c|c c c} x &0.5 &0.25 &0.25\\ \hline P(x) &-0.4 &0.6 &0.8\\ \end{array}\], \[\begin{array}{c|c c c c c} x &1.1 &2.5 &4.1 &4.6 &5.3\\ \hline P(x) &0.16 &0.14 &0.11 &0.27 &0.22\\ \end{array}\], \[\begin{array}{c|c c c c c} x &0 &1 &2 &3 &4\\ \hline P(x) &-0.25 &0.50 &0.35 &0.10 &0.30\\ \end{array}\], \[\begin{array}{c|c c c } x &1 &2 &3 \\ \hline P(x) &0.325 &0.406 &0.164 \\ \end{array}\], \[\begin{array}{c|c c c c c} x &25 &26 &27 &28 &29 \\ \hline P(x) &0.13 &0.27 &0.28 &0.18 &0.14 \\ \end{array}\]. We have to find the probability that x is between 50 and 70 or P ( 50< x < 70) For x = 50 , z = (50 - 50) / 15 = 0 For x = 70 , z = (70 - 50) / 15 = 1.33 (rounded to 2 decimal places) Consider a scenario with more than one random variable. A traveling salesman makes a sale on \(65\%\) of his calls on regular customers. If a carrier (not known to be such, of course) is boarded with three other dogs, what is the probability that at least one of the three healthy dogs will develop kennel cough? Find the probability that the proofreader will miss at least one of them. Construct the probability distribution of \(X\), the number of sales made each day. The expected value for a random variable, X, for a Bernoulli distribution is: E [X] = p. For example, if p = .04, then E [X] = 0.04. << /Type /XRef /Length 63 /Filter /FlateDecode /DecodeParms << /Columns 5 /Predictor 12 >> /W [ 1 3 1 ] /Index [ 118 14 ] /Info 19 0 R /Root 120 0 R /Size 132 /Prev 176074 /ID [<3fdbae2f5fd1eeb1cd674e4863b1705d><2fc8ffaab520ea6aadd1ebf73ff7b27f>] >> Construct the probability distribution of \(X\). 4. 0i`52>3A ZX-a6o{#IItUNAJ: DOeA>oh6{W6j`m;Pn[cU'B&B For $y \in [0,\infty)$, we have. \(X\) is the number of dice that show an even number of dots on the top face when six dice are rolled at once. Bivariate Random Variables. Solution to Example 5. a) We first calculate the mean . = f x f = 12 0 + 15 1 + 6 2 + 2 3 12 + 15 + 6 + 2 0.94. Find the average time the bus takes to drive the length of its route. When dropped on a hard surface a thumbtack lands with its sharp point touching the surface with probability \(2/3\); it lands with its sharp point directed up into the air with probability \(1/3\). Before data is collected, we regard observations as random variables (X 1,X 2,,X n) This implies that until data is collected, any function (statistic) of the observations (mean, sd, etc.) Find the expected value to the company of a single policy if a person in this risk group has a \(97.25\%\) chance of surviving one year. A Random Variable is a set of possible values from a random experiment. What is a Bernoulli Trial? \(P(Z \le 2) = P(Z \in Q = Q1M1 \bigvee Q2M2)\), where \(M1 = \{(t, u): 0 \le t \le 1, 0 \le u \le 1 + t\}\), \(M2 = \{(t, u) : 1 < t \le 2, 0 \le u \le 1 + t\}\), \(Q1 = \{(t, u) : 0 \le t \le 1/2\}\), \(Q2 = \{(t, u) : u \le 2 - t\}\) (see figure), \(P = \dfrac{3}{88} \int_{0}^{1/2} \int_{0}^{1 + t} (2t + 3u^2) du\ dt + \dfrac{3}{88} \int_{1}^{2} \int_{0}^{2 - t} (2t + 3u^2) du\ dt = \dfrac{563}{5632}\). If the ball lands in an even numbered slot, he receives back the dollar he bet plus an additional dollar. Let $X \sim Uniform(-\frac{\pi}{2},\pi)$ and $Y=\sin(X)$. The tack is dropped and its landing position observed \(15\) times. This problem has been solved! Find the expected value to the company of a single policy if a person in this risk group has a \(99.62\%\) chance of surviving one year. \end{array} \right. If each die in a pair is loaded so that one comes up half as often as it should, six comes up half again as often as it should, and the probabilities of the other faces are unaltered, then the probability distribution for the sum. Find two symmetric values "a" and "b" such that Probability [ a < X < b ] = .99 . Each of these examples contains two random variables, and our interest lies in how they are related to each other. The number of arrivals at an emergency room between midnight and 6: 00 a. m. The weight of a box of cereal labeled " 18 ounces." The duration of the next outgoing telephone call from a business office. Use the result of Exercise 10.4.2. to determine the probability \(Z \le 700, 500, 200\). Will the owner have the cover installed? Interpret the mean in the context of the problem. But if it lands tails, then we lose (failure). A coin is bent so that the probability that it lands heads up is \(2/3\). Probability models example: frozen yogurt. Discrete or Continuous Random Variables? 9.6 mobile phones.) The number of accident-free days in one month at a factory. \[\begin{array}{c|c c c c c c} x &42 &43 &44 &45 &46 &47 \\ \hline P(x) &0.10 &0.23 &0.34 &0.25 &0.05 &0.02\\ \end{array}\]. The number or bad checks drawn on Upright Bank on a day selected at random c. The amount of gasoline needed to drive your car 200 miles d. The number of traffic fatalities per year in the state of Florida e. Discrete and continuous random variables. stream The left hand side is a double integral. \begin{equation} If you'd like a pdf document containing the solutions the download tab above contains links to pdf's containing the solutions for the full book, chapter and section. Use a discrete approximation to calculate the same probablities.'. Random variables. Find the probability that such a shipment will be accepted. Examples of binomial distribution problems: The number of defective/non-defective products in a production run. Determine whether or not the random variable \(X\) is a binomial random variable. For the distributions in Exercises 10-15 below. Solution to Example 4, Problem 1 (p. 4) 0.5714 Solution to Example 4, Problem 2 (p. 5) 4 5 Glossary De nition 1: Conditional Probability The likelihood that an event will occur given that another event has already occurred. Vote counts for a candidate in an election. The distance a rental car rented on a daily rate is driven each day. $$P(X \geq \frac{1}{2})=\frac{3}{2} \int_{\frac{1}{2}}^{1} x^2dx=\frac{7}{16}.$$. One-third of all patients who undergo a non-invasive but unpleasant medical test require a sedative. The probability that an egg in a retail package is cracked or broken is \(0.025\). Suppose we flip a coin only once. A professional proofreader has a \(98\%\) chance of detecting an error in a piece of written work (other than misspellings, double words, and similar errors that are machine detected). For example, having two bowls,. \end{equation} 1 Suppose X is a nonnegative, absolutely continuous random variable. What is the mean and variance of the number of wells that must be drilled if the oil company wants to set up three producing wells? Let \(X\) denote the number of times a fair coin lands heads in three tosses. Construct the probability distribution of \(X\). \[\begin{array}{c|c c c } x &100 &101 &102 \\ \hline P(x) &0.01 &0.96 &0.03 \\ \end{array}\], Three fair dice are rolled at once. Which one? Let \(X\) denote the number of boys in a randomly selected three-child family. Determine \(P(Z \ge 1000)\), \(P(Z \ge 1300)\) and \(P(900 \le Z \le 1400)\). The mean \(\mu\) of \(X\). This is shown by the Fundamental Theorem of Calculus. Ans: Discrete d. Construct a probability distribution for this experiment. If $Y=X^2$, find the CDF of $Y$. Determine the value \(C\) must have in order for the company to average a net gain of \(\$250\) per policy on all such policies. If so, give the values of \(n\) and\(p\). RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 1. The number \(X\) of nails in a randomly selected \(1\)-pound box has the probability distribution shown. It is at the second equal sign that you can see how the general negative binomial problem reduces to a geometric random variable problem. Of all college students who are eligible to give blood, about \(18\%\) do so on a regular basis. Thus, we can use The number of boys in a randomly selected three-child family. What is the probability that the third strike comes on the seventh well drilled? A blood sample is taken from each of the individuals. 0 & \quad \text{otherwise} If the ball does not land on red he loses his dollar. Let Xand Y be two N 0-valued random variables such that X= Y+ Z, where Zis a Bernoulli random variable with parameter p2(0;1), independent of Y. We will begin with the simplest such situation, that of pairs of random variables or bivariate distributions, where we will already encounter most of the key ideas. Another example of a continuous random variable is the height of a randomly selected high school student. Two units in each shipment are selected at random and tested. \(f_{XY} (t, u) = \dfrac{24}{11}\) for \(0 \le t \le 2\), \(0 \le u \le \text{min } \{1, 2 - t\}\)(see Exercise 17 from "Problems on Random Vectors and Joint Distributions"). a) one goal in a given match. Such a person wishes to buy a \(\$75,000\) one-year term life insurance policy. \[\begin{array}{c|c c c c} x &0 &1 &2 &3 \\ \hline P(x) &0.0173 &0.0867 &0.1951 &0.2602\\ \end{array}\] \[\begin{array}{c|c c c c} x &4 &5 &6 &7 \\ \hline P(x) &0.2276 &0.1365 &0.0569 &0.0163\\ \end{array}\] \[\begin{array}{c|c c c } x &8 &9 &10 \\ \hline P(x) &0.0030 &0.0004 &0.0000 \\ \end{array}\]. Then 0 < Z C. Use properties of the exponential and natural log function to show that F Z ( v) = 1 F X ( In ( v / C) a) for 0 < v C Answer Exercise 10.4. If units remain in stock at the end of the season, they may be returned with recovery of \(r\) per unit. Problem 5. 118 0 obj \(Z = I_M (X, Y) (X + Y) + I_{M^c} (X, Y) 2Y^2\), \(M = \{(t, u): t \le 1, u \ge 1\}\), \(P(Z \le 2) = P((X, Y) \in M_1 Q_1 \bigvee (M_2 \bigvee M_3) Q_2)\), \(M_1 = \{(t, u): 0 \le t \le 1, 1 \le u \le 2\}\), \(M_2 = \{(t, u) : 0 \le t \le 1, 0 \le u \le 1\}\) \(M_3 = \{(t, u): 1 \le t \le 2, 0 \le u \le 3 - t\}\), \(Q_1 = \{(t, u): u \le 1 - t\}\) \(Q_2 = \{(t, u) : u \le 1/2\}\) (see figure), \(P = \dfrac{12}{179} \int_{0}^{1} \int_{0}^{2 - t} (3t^2 + u) du\ dt + \dfrac{12}{179} \int_{1}^{2} \int_{0}^{1} (3t^2 + u) du\ dt = \dfrac{119}{179}\), \(f_{XY} (t, u) = \dfrac{12}{227} (3t + 2tu)\), for \(0 \le t \le 2\), \(0 \le u \le \text{min } \{1 + t, 2\}\) (see Exercise 20 from "Problems on Random Variables and joint Distributions"), \(Z = I_M (X, Y) X + I_{M^c} (X, Y) \dfrac{Y}{X}\), \(M = \{(t, u): u \le \text{min } (1, 2 - t)\}\), \(P(Z \le 1) = P((X, Y) \in M_1 Q_1 \bigvee V_2Q_2)\), \(M_1 = M\), \(M_2 = M^c\), \(Q_1 = \{(t, u): 0 \le t \le \}\) \(Q_2 = \{(t, u) : u \le t\}\) (see figure), \(P = \dfrac{12}{227} \int_{0}^{1} \int_{0}^{1} (3t + 2tu) du\ dt + \dfrac{12}{227} \int_{1}^{2} \int_{2 - t}^{t} (3t + 2tu) du\ dt = \dfrac{124}{227}\). Thus, Var$\left(\frac{1}{X}\right)=E[\frac{1}{X^2}]-(E[\frac{1}{X}])^2=\frac{71}{144}$. there are two solutions to $y=g(x)$, while for $y \in (-1,0)$, there is only one solution. Denition of a Discrete Random Variable. Find the probability that two such proofreaders working independently will miss at least one error in a work that contains four errors. no: the sum of the probabilities exceeds \(1\), no: the sum of the probabilities is less than \(1\), \[\begin{array}{c|c c c c} x &0 &1 &2 &3 \\ \hline P(x) &1/8 &3/8 &3/8 &1/8\\ \end{array}\], \[\begin{array}{c|c c c c} x &-1 &999 &499 &99 \\ \hline P(x) &\frac{4987}{5000} &\frac{1}{5000} &\frac{2}{5000} &\frac{10}{5000}\\ \end{array}\], \[\begin{array}{c|c c c } x &C &C &-150,000 \\ \hline P(x) &0.9825 & &0.0175 \\ \end{array}\], \[\begin{array}{c|c c } x &-1 &1 \\ \hline P(x) &\frac{20}{38} &\frac{18}{38} \\ \end{array}\], \[\begin{array}{c|c c c c c c} x &0 &1 &2 &3 &4 &5 \\ \hline P(x) &\frac{6}{36} &\frac{10}{36} &\frac{8}{36} &\frac{6}{36} &\frac{4}{36} &\frac{2}{36} \\ \end{array}\], \[\begin{array}{c|c c c } x &0 &1 &2 \\ \hline P(x) &0.902 &0.096 &0.002 \\ \end{array}\]. The context of the problem $ and $ Y=\sin ( X ) $ and Y=\sin. 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