Chapter 1: Problem 5
Let us select five cards at random and without replacement from an ordinary deck of playing cards. (a) Find the pmf of \(X\), the number of hearts in the five cards. (b) Determine \(P(X \leq 1)\).
Chapter 1: Problem 5
Let us select five cards at random and without replacement from an ordinary deck of playing cards. (a) Find the pmf of \(X\), the number of hearts in the five cards. (b) Determine \(P(X \leq 1)\).
All the tools & learning materials you need for study success - in one app.
Get started for freeEach 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?
A mode of a distribution of one random variable \(X\) is a value of \(x\) that
maximizes the pdf or pmf. For \(X\) of the continuous type, \(f(x)\) must be
continuous. If there is only one such \(x\), it is called the mode of the
distribution. Find the mode of each of the following distributions:
(a) \(p(x)=\left(\frac{1}{2}\right)^{x}, x=1,2,3, \ldots\), zero elsewhere.
(b) \(f(x)=12 x^{2}(1-x), 0
Let \(X\) be a random variable of the continuous type that has pdf \(f(x)\). If \(m\) is the unique median of the distribution of \(X\) and \(b\) is a real constant, show that $$E(|X-b|)=E(|X-m|)+2 \int_{m}^{b}(b-x) f(x) d x$$ provided that the expectations exist. For what value of \(b\) is \(E(|X-b|)\) a minimum?
Consider \(k\) continuous-type distributions with the following characteristics: pdf \(f_{i}(x)\), mean \(\mu_{i}\), and variance \(\sigma_{i}^{2}, i=1,2, \ldots, k .\) If \(c_{i} \geq 0, i=1,2, \ldots, k\), and \(c_{1}+c_{2}+\cdots+c_{k}=1\), show that the mean and the variance of the distribution having pdf \(c_{1} f_{1}(x)+\cdots+c_{k} f_{k}(x)\) are \(\mu=\sum_{i=1}^{k} c_{i} \mu_{i}\) and \(\sigma^{2}=\sum_{i=1}^{k} c_{i}\left[\sigma_{i}^{2}+\left(\mu_{i}-\mu\right)^{2}\right]\) respectively.
From a bowl containing 5 red, 3 white, and 7 blue chips, select 4 at random and without replacement. Compute the conditional probability of 1 red, 0 white, and 3 blue chips, given that there are at least 3 blue chips in this sample of 4 chips.
What do you think about this solution?
We value your feedback to improve our textbook solutions.