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Chapter 3: Problem 5

Determine the value \(c\) so that each of the following functions can serve as a probability distribution of the discrete random variable \(X\) : (a) \(f(x)=c\left(x^{2}+4\right),\) for \(a:=0,1,2,3\) (b) \(f(x)=c\left(\begin{array}{c}2 \\\ x\end{array}\right)\left(\begin{array}{c}3 \\ 3-x\end{array}\right),\) for \(z=0,1,2\)

Short Answer

Expert verified
The value of \(c\) for function (a) and function (b) are determined by setting up and solving equations based on the requirement that the sum of all probabilities equals 1. After setting up these equations using the given functions and solving them, we find the desired values for \(c\).

Step by step solution

01

Determine the 'c' for the function in Part (a)

The function for a is \(f(x) = c(x^2 + 4)\) for \(x=0,1,2,3\). As it is a probability distribution, we know that the sum of the probabilities for all possible outcomes equals 1. Using this information, we can set up the following equation: \(c(0^2 + 4) + c(1^2 + 4) + c(2^2 + 4) + c(3^2 + 4) = 1\). Simplifying and solving this equation will give us the value for \(c\).
02

Determine the 'c' for the function in Part (b)

The function for b is \(f(x) = c\left(\binom{2}{x}\right)\left(\binom{3}{3-x}\right)\) for \(x=0,1,2\). The binomial coefficients symbolize the number of ways to choose \(x\) items from a set of 2 items and the number of ways to choose \(3-x\) items from a set of 3 items. Similar to Step 1, we can set up the following equation: \(c\left(\binom{2}{0}\right)\left(\binom{3}{3}\right) + c\left(\binom{2}{1}\right)\left(\binom{3}{2}\right) + c\left(\binom{2}{2}\right)\left(\binom{3}{1}\right) = 1\). Simplifying and solving this equation will give us the value for \(c\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Random Variable
A discrete random variable is a type of variable used in probability to represent countable outcomes of an experiment or random process. This simply means that the variable has distinct and separate values, like the number of coins in a jar or the roll of a die. They contrast with continuous random variables, which can take on any value within a range.

In the exercise provided, the random variable X can take on the values 0, 1, 2, and 3 (part a) or 0, 1, and 2 (part b). This clearly fits the definition of a discrete random variable because X is limited to specific, countable values. Understanding this concept helps ensure that the correct methods are used to determine the probabilities associated with X.
Binomial Coefficients
Binomial coefficients are an essential concept in combinatorics and represent the number of ways you can choose a subset of items from a larger set without regard to the order in which they are chosen. Notationally, they are represented as \(\binom{n}{k}\), which equals the number of ways to pick k items from n items, and can be calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) where '!' denotes factorial.

In part (b) of the exercise, the binomial coefficients are used within the probability distribution function, and they play a vital role in calculating the value of 'c'. Understanding binomial coefficients is crucial since they form the basis of the binomial probability distribution. They are a practical concept for calculating certain types of probability where fixed numbers of trials and successes are involved.
Probability Distribution Properties
A probability distribution is a mathematical description of the likelihood of occurrence of different possible outcomes in an experiment. For discrete random variables, the probability distribution is often represented in the form of a table, formula, or graph. There are several properties that these distributions always follow:
  • The probability of each outcome in the distribution must be between 0 and 1, inclusive.
  • The sum of the probabilities of all the possible outcomes must equal 1.
  • If two events are mutually exclusive, the probability of either occurring is the sum of their individual probabilities.
In the given exercise, the function f(x), whether part (a) or part (b), describes a probability distribution for the random variable X. To confirm that these functions can be valid probability distributions, you must find the value of 'c' that satisfies these properties鈥攆undamentally ensuring the resulting probabilities sum to 1, as noted in the steps provided.

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Most popular questions from this chapter

Classify the following random variables as discrete or continuous: \(X\) : the number of automobile accidents per year in Virginia. \(Y:\) the length of time to play 18 holes of golf. \(M\) : the amount of milk produced yearly by a particular cow. \(N:\) the number of eggs laid each month by a hen. \(P:\) the number of building permits issued each month in a certain city. \(Q:\) the weight of grain produced per acre.

The total number of hours, measured in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is a continuous random variable \(X\) that has the density function $$ f(x)=\left\\{\begin{array}{ll} x_{1} & 0 < x< 1, \\ 2-x, & 1 \leq x <2, \\ 0, & \text { elsewhere } \end{array}\right. $$ Find the probability that over a period of one year, a family runs their vacuum cleaner (a) less than 120 hours; (b) between 50 and 100 hours,

Given the joint density function $$ f(x, \quad y)=\left\\{\begin{array}{ll} \frac{6-x-y}{8} ; & 0< x <2,2 < y <4\. \\ 0, & \text { elsewhere. } \end{array}\right. $$

Suppose a special type of small data processing firm is so specialized that some have difficulty making a profit in their first year of operation. The pdf that characterizes the proportion \(Y\) that make a profit is given by $$ f(x)=\left\\{\begin{array}{ll} k y^{4}(1-y)^{3}+ & 0 \leq y \leq 1, \\ 0_{1} & \text { elsewhere. } \end{array}\right. $$ (a) What is the value of \(k\) that renders the above a valid density function? (b) Find the probability that at most \(50 \%\) of the firms make a profit in the first year. (c) Find the probability that at least \(80 \%\) of the firms make a profit in the first year.

Pairs of pants are being produced by a particular outlet facility. The pants are "checked" by a group of 10 workers. The workers inspect pairs of pants taken randomly from the production line. Each inspector is assigned a number from 1 through \(10 .\) A buyer selects a pair of pants for purchase. Let the random variable \(X\) be the inspector number. (a) Give a reasonable probability mass function for \(X\). (b) Plot the cumulative distribution function for \(X\).
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