Show that the probability that any such mixture will contain the blood of at least one diseased person, hence test positive, is about \(0.18\). Lesson 20: Distributions of Two Continuous Random Variables, 20.2 - Conditional Distributions for Continuous Random Variables, Lesson 21: Bivariate Normal Distributions, 21.1 - Conditional Distribution of Y Given X, Section 5: Distributions of Functions of Random Variables, Lesson 22: Functions of One Random Variable, Lesson 23: Transformations of Two Random Variables, Lesson 24: Several Independent Random Variables, 24.2 - Expectations of Functions of Independent Random Variables, 24.3 - Mean and Variance of Linear Combinations, Lesson 25: The Moment-Generating Function Technique, 25.3 - Sums of Chi-Square Random Variables, Lesson 26: Random Functions Associated with Normal Distributions, 26.1 - Sums of Independent Normal Random Variables, 26.2 - Sampling Distribution of Sample Mean, 26.3 - Sampling Distribution of Sample Variance, Lesson 28: Approximations for Discrete Distributions, Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris, Duis aute irure dolor in reprehenderit in voluptate, Excepteur sint occaecat cupidatat non proident. The coin could travel 1 cm, or 1.1 cm, or 1.11 cm, or on and on. Identify the set of possible values for each random variable. Except where otherwise noted, content on this site is licensed under a CC BY-NC 4.0 license. The left hand side is a double integral. Let x be the random variable that represents the length of time. The distance a rental car rented on a daily rate is driven each day. This module concerns discrete random variables. Minterm probabilities are (in the usual order), \(Y = I_D + 3I_E + I_F - 3\). Find $P(X \leq \frac{2}{3} | X> \frac{1}{3})$. a) one goal in a given match. Define a new random variable as Z = X + Y. \end{equation} The'correlation'coefficient'isa'measure'of'the' linear$ relationship between X and Y,'and'onlywhen'the'two' variablesare'perfectlyrelated'in'a'linear'manner'will' be 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? Interpret the mean in the context of the problem. Discrete and continuous random variables. The set of possible values X can take on is its range . This contains answers about the probability worksheet. Verify that \(X\) satisfies the conditions for a binomial random variable, and find \(n\) and \(p\). The number of vehicles owned by a randomly selected household. Based on the result in (b), show that the expected number of mixtures that test positive is about \(11\). % Construct the probability distribution of \(X\). "Introductory Statistics" by Shafer and Zhang. For the pair \(\{X, Y\}\) in Exercise 8, let. Such a person wishes to buy a \(\$150,000\) one-year term life insurance policy. Find two symmetric values "a" and "b" such that Probability [ a < X < b ] = .99 . let the random variable, be the number . In particular, it is the integral of $f_X(t)$ over \(X\) is a binomial random variable with parameters \(n=16\) and \(p=0.74\). Fig.4.4 - The shaded area shows the region of the double integral of Problem 5. First, -1 indicates a perfect inverse relationship (i.e., a unit change in one means that the other will have a unit change in the opposite direction). We have In a \(\$1\) bet on red, the bettor pays \(\$1\) to play. 0 & \quad \text{otherwise} An insurance company will sell a \(\$10,000\) one-year term life insurance policy to an individual in a particular risk group for a premium of \(\$368\). the shaded region in Figure 4.4. A laboratory performs \(20\) such tests daily. 9.6 mobile phones.) Arcu felis bibendum ut tristique et egestas quis: An oil company conducts a geological study that indicates that an exploratory oil well should have a 20% chance of striking oil. \(X\) is the number of times the number of dots on the top face of a fair die is even in six rolls of the die. \2013\PubHlth 540 Word Problems Unit 5.doc Solution Using Z-Score: Step 1 Launch the David Lane normal distribution calculator provided to you on the topic page (5. That distance, x, would be a continuous random variable because it could take on a infinite number of values within the continuous range of real numbers. /OcU>x k-kM[;AvrBI'JUf&X4\c$s!- 'eww:~wH]m6_,jw)eyUUwQ++"^"m[/X5K\ au; AP~- ^^@omrRH+&%"< wm=-PTXY/WPw\?piE*v{nnX#CfncR M`b0U&M}1)}Eh0E{Mf|da.jL %bhjK%LH)^)mrR3-k M fqIX(;D@73eJ Find the probability that \(X\) is at least five. The tourist sees four local people standing at a bus stop. By contrast, a continuous random variable can take any value, in principle, within a specied range. Find the average number of inferior quality grapefruit per box of a dozen. Determine \(P(\text{max }\{X, Y\} \le 4)\). This is shown by the Fundamental Theorem of Calculus. every payday, at which time there are always two tellers on duty. For $y \in [0,\infty)$, we have. The number \(X\) of customers in the bank who are either at a teller window or are waiting in a single line for the next available teller has the following probability distribution. If the player rolls doubles all three times there is a penalty. Experience indicates demand can be represented by a random variable \(D\) ~ Poisson (\(\mu\)). (1) Discrete random variable. Thus, we need to show that Examples of Textual Aids; Newest. \(X\) is a binomial random variable with the parameters shown. The weight of a box of cereal labeled \(18\) ounces.. (For fair dice this number is \(7\)). At this time, I do not offer pdf's for solutions to . Let \(X\) denote the net gain from the purchase of a randomly selected ticket. . This page titled 10.4: Problems on Functions of Random Variables is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Paul Pfeiffer via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. If so, give the values of \(n\) and\(p\). The value of X can be 68, 71.5, 80.6, or 90.32. 119 0 obj If the cost of replacement at failure is \(C\) dollars, then the present value of the replacement is \(Z = Ce^{-aX}\). We start with the de nition a continuous random ariable.v De nition (Continuous random ariabvles) A random arviable Xis said to have a ontinuousc distribution if there exists a non-negative function f= f X such that P(a6X6b) = b a f(x)dx for every aand b. A random variable based on a count is an example of a discrete random variable. De nition, PDF, CDF. Consider the following experiment. Ten percent of all purchasers of a refrigerator buy an extended warranty. Solution The ve residuals become: -1.4, 2.6, -1.4, 0.6 og -0.4. A Bernoulli random variable has the following properties: Bernoulli Distribution Mean And Variance Worked Example Let's look at an example of a Bernoulli random variable. The time, to the nearest whole minute, that a city bus takes to go from one end of its route to the other has the probability distribution shown. Now, I would understand if you feel, "Why should we learn to do the condence . xrOvSxLNU&9$$"CG 9Hh<4oiqS=2OV^hn+]\P"U W|qwsElL mP'-/^)S# Find $f_Y(y)$. It is at the second equal sign that you can see how the general negative binomial problem reduces to a geometric random variable problem. What is the probability that the third strike comes on the seventh well drilled? If not, explain why not. Classify each random variable as either discrete or continuous. Example: Tossing a coin: we could get Heads or Tails. The probability of rolling doubles in a single roll of a pair of fair dice is \(1/6\). Use the Central Limit Theorem (applied to a negative binomial random variable) to estimate the probability that more than 50 tosses are needed. In a random sample of \(20\) adults, \(14\) recognized its brand name. Assuming that boys and girls are equally likely, construct the probability distribution of \(X\). For example, the height of students in a class, the amount of ice tea in a glass, the change in temperature throughout a day, and the number of hours a person works in a week all contain a range of values in an interval, thus continuous random variables. \(P(Z \le v) = (\dfrac{v}{1000})^{10/7}\), Optimal stocking of merchandise. Let X = the number of defective bulbs selected. A Quality Control Inspector randomly samples 4 bulbs without replacement. The air pressure of a tire on an automobile. Chapter 3 : Derivatives. The module Continuous probability If $y \in (-1,0)$ we have one solution, $x_1=\arcsin(y)$. A bivariate distribution, put simply, is the probability that a certain event will occur when there are two independent random variables in your scenario. Its set of possible values is the set of real numbers R, one interval, or a disjoint union of intervals on the real line (e.g., [0, 10] [20, 30]). By looking at An insurance company estimates that the probability that an individual in a particular risk group will survive one year is \(0.9825\). The function fis called the density function for Xor the PDF . Answer The key to finding c is to use item #2 in the definition of a p.m.f. Find the probability that it lands heads up at most five times. 122 0 obj These values are obtained by measuring by a thermometer. Next, run a computer simulation to carry out this experiment. For concreteness, start with two, . random variable) to estimate the probability that fewer than 20 of those tosses come up heads. Explain fully. Definition The binomial random variable X associated with a binomial experiment consisting of n trials is defined as X = the number of S's among the n trials Let \(Z = 3X^3 + 3X^2 Y - Y^3\). 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. About \(2\%\) of alumni give money upon receiving a solicitation from the college or university from which they graduated. \begin{equation} Note that \(X\)is technically a geometric random variable, since we are only looking for one success. If X N (0, 1), how many realizations out of 10000 realizations of X do you expect to be between 1 and 3 ? If the ball lands in an even numbered slot, he receives back the dollar he bet plus an additional dollar. Six men and ve women apply for an executive position in a small company. For example, suppose an experiment is to measure the arrivals of cars at a tollbooth during a minute period. Adverse growing conditions have caused \(5\%\) of grapefruit grown in a certain region to be of inferior quality. Then one dollar in hand now, has a value \(e^{ax}\) at the end of \(x\) years. \(X\) is a binomial random variable with the parameters shown. That is, there is about a 5% chance that the third strike comes on the seventh well drilled. A random variable (rv) is a numeric function of the outcome, X : !R. A random variable X is said to be discrete if it can assume only a nite or countable innite number of distinct values. Determine whether or not the random variable \(X\) is a binomial random variable. Find the average number of nails per pound. Probability models. be divided to a finite number of regions in which it is monotone. Additional tickets are available according to the following schedule: If the number of purchasers is a random variable \(X\), the total cost (in dollars) is a random quantity \(Z = g(X)\) described by, \(g(X) = 200 + 18 I_{M1} (X) (X - 10) + (16 - 18) I_{M2} (X) (X - 20) +\), \((15 - 16) I_{M_3} (X) (X - 30) + (13 - 15) I_{M4} (X) (X - 50)\), where \(M1 = [10, \infty)\), \(M2 = [20, \infty)\), \(M3 = [30, \infty)\), \(M4 = [50, \infty)\). \(P(Z \le 2) = P(Z \in Q = Q1M1 \bigvee Q2M2)\), 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), \(P = \dfrac{3}{88} \int_{0}^{1/2} \int_{0}^{1 + t} (2t + 3u^2) du\ dt + \dfrac{3}{88} \int_{1}^{2} \int_{0}^{2 - t} (2t + 3u^2) du\ dt = \dfrac{563}{5632}\). Ans: Discrete d. Construct a probability distribution for this experiment. Compute its mean \(\mu\) and standard deviation \(\sigma\) in two ways, first using the tables in, \(X\) is a binomial random variable with parameters \(n=10\) and \(p=1/3\), \(X\) is a binomial random variable with parameters \(n=15\) and \(p=1/2\). To do so assume that if the cover were in place the revenue each night of the season would be the same as the revenue on a clear night. Let Xi denote the number of times that outcome Oi occurs in the n repetitions of the experiment. Determine and plot the distribution function for \(W\). \(X\) is the number of dots on the top face of fair die that is rolled. Find the average number monetary gifts a college can expect from every \(2,000\) solicitations it sends. Suppose we flip a coin only once. Examples of binomial distribution problems: The number of defective/non-defective products in a production run. 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) Approximate the Poisson distribution by truncating at 150. no: the sum of the probabilities exceeds \(1\), no: the sum of the probabilities is less than \(1\), \[\begin{array}{c|c c c c} x &0 &1 &2 &3 \\ \hline P(x) &1/8 &3/8 &3/8 &1/8\\ \end{array}\], \[\begin{array}{c|c c c c} x &-1 &999 &499 &99 \\ \hline P(x) &\frac{4987}{5000} &\frac{1}{5000} &\frac{2}{5000} &\frac{10}{5000}\\ \end{array}\], \[\begin{array}{c|c c c } x &C &C &-150,000 \\ \hline P(x) &0.9825 & &0.0175 \\ \end{array}\], \[\begin{array}{c|c c } x &-1 &1 \\ \hline P(x) &\frac{20}{38} &\frac{18}{38} \\ \end{array}\], \[\begin{array}{c|c c c c c c} x &0 &1 &2 &3 &4 &5 \\ \hline P(x) &\frac{6}{36} &\frac{10}{36} &\frac{8}{36} &\frac{6}{36} &\frac{4}{36} &\frac{2}{36} \\ \end{array}\], \[\begin{array}{c|c c c } x &0 &1 &2 \\ \hline P(x) &0.902 &0.096 &0.002 \\ \end{array}\]. 4.1 Mean of a Random Variable The expected value, or mathematical expectation E(X) of a random variable X is the long-run average value of X . 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? Truncate \(X\) at 1000 and use 10,000 approximation points. \nonumber f_X(x) = \left\{ Although $g$ is not monotone, it can A random variable X has a Bernoulli distribution with parameter p, where 0 p 1, if it has only two possible values, typically denoted 0 and 1. \(f_{XY} (t, u) = \dfrac{3}{23} (t + 2u)\) for \(0 \le t \le 2\), \(0 \le u \le \text{max } \{2 - t, t\}\) (see Exercise 18 from "Problems on Random Vectors and Joint Distributions"). In most practical problems: o A discrete random variable represents count data, such as the number of defectives in a sample of k items. \(X\) is the number of black marbles in a sample of \(5\) marbles drawn randomly and without replacement from a box that contains \(25\) white marbles and \(15\) black marbles. (Make a reasonable estimate based on experience, where necessary.). The class \(\{X, Y, Z\}\) is independent. Suppose \(\lambda = 1/10\), \(a = 0.07\), and \(C =\) $1000. The number of accident-free days in one month at a factory. Use the answer to (a) to compute the projected total revenue per \(90\)-night season if the cover is not installed. there are two solutions to $y=g(x)$, while for $y \in (-1,0)$, there is only one solution. A fair coin is tossed repeatedly until either it lands heads or a total of five tosses have been made, whichever comes first. Equation 4.6. Using the answers to (b) and (c), decide whether or not the additional cost of the installation of the cover will be recovered from the increased revenue over the first ten years. 2. That is, the random variables Xand Yhave the same distribution, but the random vectors (X;Y) and (Y;X) don't. (d) Sampling questions revisited The independent events A0 i from example (a) are exchangeable, because of formula (1). \end{equation}. Let $X$ be a positive continuous random variable. 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). 4. For the pair \(\{X, Y\}\) in Exercise 10.4.8, let \(Z = g(X, Y) = 3X^2 + 2XY - Y^2\). Grapefruit are sold by the dozen. Yes/No Survey (such as asking 150 people if they watch ABC news). Creative Commons Attribution NonCommercial License 4.0. X is an example of a random variable, which brings us to the following de nition: De nition 3.1.1: Random Variable Suppose we conduct an experiment with sample space . Solution Problem (a) X+ Zand Y+ Zare independent (b) Xhas to be 2N 0 = f0;2;4;6;:::g-valued We have already seen examples of continuous random variables, when the idea of a ran-dom variable was rst introduced. The number of games in the next World Series (best of up to seven games). Let \(X\) denote the net gain to the bettor on one play of the game. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Other examples of continuous random variables would be the mass of stars in our galaxy, the pH of ocean waters, or the . If demand exceeds the number originally ordered, extra units may be ordered at a cost of s each. endobj -*A @f 46 $$f_X(x)=\frac{1}{2}e^{-|x|}, \hspace{20pt} \textrm{for all }x \in \mathbb{R}.$$ Plot the distribution function \(F_X\) and the quantile function \(Q_X\). A box that contains two or more grapefruit of inferior quality will cause a strong adverse customer reaction. Is the random variable, x, continuous or discrete? Two units in each shipment are selected at random and tested. A coin is bent so that the probability that it lands heads up is \(2/3\). Example: problem 5.1: y p(x;y) 0 1 2 0 :10 :04 :02 x 1 :08 :20 :06 . Find the probability that it lands heads up more times than it lands tails up. (b) Hence find the expectation and variance of X. The sample is large enough: n = 40( 30). Compute the mean revenue per night if the cover is not installed. is also a random variable Thus, any statistic, because it is a random variable, has a probability distribution - referred to as a sampling . Secondly, +1 indicates a perfect linear relationship (i.e., the two variables move in the same direction with equal unit changes). Present value of future costs. Find the probability that no days at all will be lost next summer. (See Example 2 from "Functions of a Random Variable") The cultural committee of a student organization has arranged a special deal for tickets to a concert. Normal) of 123 0 obj Rather it is a weighted average of the possible values. Seven thousand lottery tickets are sold for \(\$5\) each. laudantium assumenda nam eaque, excepturi, soluta, perspiciatis cupiditate sapiente, adipisci quaerat odio (Supposing that indeed \(11\) of the \(60\) mixtures test positive, then we know that none of the \(490\) persons whose blood was in the remaining \(49\) samples that tested negative has the disease. voluptate repellendus blanditiis veritatis ducimus ad ipsa quisquam, commodi vel necessitatibus, harum quos The random variable X has the following PDF: f_X (x) = {1/3 -1 < x < 2 0 otherwise If we define Y = 2X + 3, what is the PDF of Y? A blood sample is taken from each of the individuals. << /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>] >> Which one? The owner of a proposed outdoor theater must decide whether to include a cover that will allow shows to be performed in all weather conditions. Let $X$ be a continuous random variable with PDF given by To find the requested probability, we need to find \(P(X=7\), which can be readily found using the p.m.f. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. A student guesses the answer to every question. endobj Let \(C\) denote how much the insurance company charges such a person for such a policy. The amount of liquid in a \(12\)-ounce can of soft drink. Let \(X\) denote the number of times a fair coin lands heads in three tosses. Thus, for $y \in(-1,0)$, we have. He needs to ask someone directions. << /Annots [ 41 0 R 42 0 R ] /Contents 123 0 R /MediaBox [ 0 0 612 792 ] /Parent 55 0 R /Resources 43 0 R /Type /Page >> Will the owner have the cover installed? If $Y=\frac{2}{X}+3$, find Var$(Y)$. Determine \(P(\{X + Y \ge 5\} \cup \{Y \le 2\})\), \(P(X^2 + Y^2 \le 10)\). $$P(X \geq \frac{1}{2})=\frac{3}{2} \int_{\frac{1}{2}}^{1} x^2dx=\frac{7}{16}.$$. What is a random variable and its types? $=\int_{-\sqrt{y}}^{\sqrt{y}} \frac{1}{2}e^{-|x|} dx$, $P(X \leq \frac{2}{3} | X > \frac{1}{3})$, $=\frac{P(\frac{1}{3} < X \leq \frac{2}{3})}{P(X > \frac{1}{3})}$, $=\frac{\int_{\frac{1}{3}}^{\frac{2}{3}} 4x^3 dx}{\int_{\frac{1}{3}}^{1} 4x^3 dx}$, $\int_{0}^{\infty} \int_{x}^{\infty}f_X(t)dtdx$, $=\int_{0}^{\infty} \int_{0}^{t}f_X(t)dx dt$, $=\int_{0}^{\infty} f_X(t) \left(\int_{0}^{t} 1 dx \right) dt$, $=\int_{0}^{\infty} tf_X(t) dt=EX \hspace{20pt} \textrm{since $X$ is a positive random variable}.$, $= \frac{f_X(\arcsin(y))}{|\cos(\arcsin(y))|}$, $= \frac{\frac{2}{3 \pi}}{\sqrt{1-y^2}}.$, $= \frac{f_X(x_1)}{|g'(x_1)|}+\frac{f_X(x_2)}{|g'(x_2)|}$, $= \frac{f_X(\arcsin(y))}{|\cos(\arcsin(y))|}+\frac{f_X(\pi-\arcsin(y))}{|\cos(\pi-\arcsin(y))|}$, $= \frac{\frac{2}{3 \pi}}{\sqrt{1-y^2}}+\frac{\frac{2}{3 \pi}}{\sqrt{1-y^2}}$. For a general bivariate case we write this as P(X 1 = x 1, X 2 = x 2). Another example of a continuous random variable is the height of a randomly selected high school student. A random variable is a variable that denotes the outcomes of a chance experiment. 0 & \quad \text{otherwise} Expected value is a summary statistic, providing a measure of the location or central tendency of a random variable. Thirty-six slots are numbered from \(1\) to \(36\); the remaining two slots are numbered \(0\) and \(00\). endstream 120 0 obj the plot of $g(x)=\sin(x)$ over $[-\frac{\pi}{2},\pi]$, we notice that for $y \in (0,1)$ Q 5.2.1. Well, I'm not sure. The pattern evident from parts (a) and (b) is that if. Find the probability that a carton of one dozen eggs contains no eggs that are either cracked or broken. Discrete or Continuous Random Variables? Random variables. { "10.01:_Functions_of_a_Random_Variable" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
random variable example problems with solutions pdf
random variable example problems with solutions pdf
random variable example problems with solutions pdf