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Chapter 1: Problem 3

Let the subsets \(C_{1}=\left\\{\frac{1}{4}

Short Answer

Expert verified
\$P_{X}\left(C_{1} \cup C_{2}\right) = 5/8$, $P_{X}\left(C_{1}^{c}\right) = 7/8$, and $P_{X}\left(C_{1}^{c} \cap C_{2}^{c}\right) = 3/8$

Step by step solution

01

Calculate \(P_{X}\left(C_{1} \cup C_{2}\right)\)

Since subsets \(C_1\) and \(C_2\) are disjoint (non-overlapping), the probability of their union is the sum of their individual probabilities. Thus, \(P_{X}\left(C_{1} \cup C_{2}\right) = P_{X}(C_1) + P_{X}(C_2) = 1/8 + 1/2 = 5/8.\
02

Calculate \(P_{X}\left(C_{1}^{c}\right)\)

The complement of a set (denoted by \(^c\) ) is the event that the set does not occur. Therefore, the probability of the complement of \(C_1\) is 1 minus the probability of \(C_1\). Thus, \(P_{X}\left(C_{1}^{c}\right) = 1 - P_{X}(C_1) = 1 - 1/8 = 7/8\
03

Calculate \(P_{X}\left(C_{1}^{c} \cap C_{2}^{c}\right)\)

The intersection of the complements of sets \(C_1\) and \(C_2\) is the set of all outcomes that are neither in \(C_1\) nor in \(C_2\). This is equivalent to the complement of the union of \(C_1\) and \(C_2\). Therefore, its probability is 1 minus the probability of the union of \(C_1\) and \(C_2\) (which was calculated in Step 1). Thus, \(P_{X}\left(C_{1}^{c} \cap C_{2}^{c}\right) = 1 - P_{X}\left(C_{1} \cup C_{2}\right) = 1 - 5/8 = 3/8\

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Most popular questions from this chapter

For each of the following cdfs \(F(x)\), find the pdf \(f(x)[\mathrm{pmf}\) in part \((\mathrm{d})]\), the 25 th percentile, and the 60 th percentile. Also, sketch the graphs of \(f(x)\) and \(F(x)\). (a) \(F(x)=\left(1+e^{-x}\right)^{-1},-\infty

Each bag in a large box contains 25 tulip bulbs. It is known that \(60 \%\) of the bags contain bulbs for 5 red and 20 yellow tulips, while the remaining \(40 \%\) of the bags contain bulbs for 15 red and 10 yellow tulips. A bag is selected at random and a bulb taken at random from this bag is planted. (a) What is the probability that it will be a yellow tulip? (b) Given that it is yellow, what is the conditional probability it comes from a bag that contained 5 red and 20 yellow bulbs?

Let \(X\) have the cdf \(F(x)\) that is a mixture of the continuous and discrete types, namely $$ F(x)=\left\\{\begin{array}{ll} 0 & x<0 \\ \frac{x+1}{4} & 0 \leq x<1 \\ 1 & 1 \leq x \end{array}\right. $$ Determine reasonable definitions of \(\mu=E(X)\) and \(\sigma^{2}=\operatorname{var}(X)\) and compute each. Hint: Determine the parts of the pmf and the pdf associated with each of the discrete and continuous parts, and then sum for the discrete part and integrate for the continuous part.

Let \(X\) be a random variable with a pdf \(f(x)\) and \(\operatorname{mgf} M(t)\). Suppose \(f\) is symmetric about \(0 ;\) i.e., \(f(-x)=f(x)\). Show that \(M(-t)=M(t)\).

For every one-dimensional set \(C\), define the function \(Q(C)=\sum_{C} f(x)\), where \(f(x)=\left(\frac{2}{3}\right)\left(\frac{1}{3}\right)^{x}, x=0,1,2, \ldots\), zero elsewhere. If \(C_{1}=\\{x: x=0,1,2,3\\}\) and \(C_{2}=\\{x: x=0,1,2, \ldots\\}\), find \(Q\left(C_{1}\right)\) and \(Q\left(C_{2}\right)\). Hint: Recall that \(S_{n}=a+a r+\cdots+a r^{n-1}=a\left(1-r^{n}\right) /(1-r)\) and, hence, it follows that \(\lim _{n \rightarrow \infty} S_{n}=a /(1-r)\) provided that \(|r|<1\).
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