answer:To find the probability density function (pdf) of the random variable ( Y = X^2 ), given the pdf of ( X ), we proceed as follows: 1. **Determine the range of ( X ) and ( Y )**: - The given pdf for ( X ) is: [ p(x) = begin{cases} 1 & text{for } 1 leq x leq 2 0 & text{for } x < 1 text{ or } x > 2 end{cases} ] - Since ( Y = X^2 ), we need to find the range of ( Y ) given the range of ( X ). - When ( X = 1 ), ( Y = 1^2 = 1 ) - When ( X = 2 ), ( Y = 2^2 = 4 ) - Thus, ( Y ) ranges from 1 to 4, i.e., ( 1 leq Y leq 4 ). 2. **Function transformation and its invertibility**: - The function ( Y = X^2 ) is monotonically increasing on ( 1 leq x leq 2 ). - The inverse function is ( x = sqrt{y} ). 3. **Determine the derivative of the inverse function**: - The inverse of ( y = x^2 ) is ( x = sqrt{y} ). - The derivative of the inverse function ( x = sqrt{y} ) with respect to ( y ) is: [ frac{dx}{dy} = frac{1}{2sqrt{y}} ] 4. **Apply the formula for the pdf of ( Y )**: - We utilize the formula for transforming the pdf under a monotonic transformation: [ g(y) = p(x) left| frac{dx}{dy} right| ] - Since ( p(x) = 1 ) for ( 1 leq x leq 2 ), substituting ( x = sqrt{y} ), we get: [ g(y) = 1 cdot left| frac{1}{2sqrt{y}} right| = frac{1}{2sqrt{y}} ] 5. **Define ( g(y) ) over the range of ( Y )**: - For ( 1 leq y leq 4 ): [ g(y) = frac{1}{2sqrt{y}} ] - For ( y < 1 ) or ( y > 4 ), the density is 0. 6. **Conclusion**: - The pdf of ( Y = X^2 ) is: [ g(y) = begin{cases} frac{1}{2sqrt{y}} & text{for } 1 leq y leq 4 0 & text{for } y < 1 text{ or } y > 4 end{cases} ] [ boxed{ g(y) = begin{cases} frac{1}{2sqrt{y}} & text{for } 1 leq y leq 4 0 & text{for } y < 1 text{ or } y > 4 end{cases} } ]

answer:Let's denote the percentage of votes the first candidate got as P%. Since the total number of votes is 1200, the number of votes the first candidate got can be represented as (P/100) * 1200. The second candidate got 480 votes. Therefore, the first candidate must have gotten the remaining votes out of the total 1200 votes. So we can write the equation: (P/100) * 1200 + 480 = 1200 Now, let's solve for P: (P/100) * 1200 = 1200 - 480 (P/100) * 1200 = 720 Now, divide both sides by 1200 to solve for P: P/100 = 720 / 1200 Simplify the fraction: P/100 = 3 / 5 Convert the fraction to a percentage: P = (3 / 5) * 100 P = 60% Therefore, the first candidate got boxed{60%} of the votes.

answer:Developing a business plan for your online handmade or crafted items business involves several key steps. Let's break down the process into manageable parts. First, define your niche by identifying the type of handmade or crafted items you want to sell. What are your skills and interests? What types of products do you enjoy making? Who is your target audience? Consider what types of items are in demand and what gaps you can fill in the market. For example, you might specialize in custom jewelry, hand-painted home decor, or artisanal candles. Next, conduct market research to understand your competition and target audience. Look at existing online businesses selling similar products and analyze their strengths, weaknesses, and pricing strategies. Identify your unique selling points and what sets your products apart. You can also gather information from social media, online forums, and customer reviews to gain insights into consumer preferences and trends. Now, outline your business model, including your mission statement, business structure, and revenue streams. Will you operate as a sole proprietorship or register your business as an LLC? How will you price your products, and what payment methods will you accept? Consider the costs of materials, shipping, and marketing when determining your pricing strategy. Develop an operations plan, detailing how you will manage the day-to-day tasks of your business. This includes sourcing materials, creating products, taking product photos, and handling customer service. You'll also need to consider how you'll manage inventory, fulfill orders, and ship products. Create a marketing strategy that outlines how you'll reach your target audience and promote your products. This may include social media marketing, email marketing, paid advertising, or influencer partnerships. Develop a content calendar to ensure consistent branding and messaging across all channels. Lastly, establish financial projections and set realistic goals for your business. Estimate your startup costs, including initial inventory, website development, and marketing expenses. Project your monthly expenses and revenue, and set milestones for growth and expansion. By considering these key areas, you'll be able to create a comprehensive business plan that guides your online handmade or crafted items business toward success. What specific area would you like to focus on next?

answer:When choosing an e-commerce platform to sell your handmade items, consider factors such as ease of use, customization options, payment processing fees, and integrations with social media or other marketplaces. Here are some popular options to consider: You can use a self-hosted e-commerce platform like Shopify, WooCommerce, or BigCommerce. These platforms offer a range of customizable templates, payment gateways, and shipping integrations. They also provide flexibility in managing your store and products. Shopify, for example, offers a user-friendly interface and a vast app store for integrations with third-party services. WooCommerce is a popular choice for those already familiar with WordPress, as it integrates seamlessly with the content management system. Another option is to sell through a marketplace platform like Etsy, eBay, or Handmade at Amazon. These platforms cater specifically to handmade or unique items and offer existing customer bases. Etsy, for example, is a well-established marketplace for handmade, vintage, and unique items. While you'll pay listing fees and commission on sales, you'll benefit from the platform's existing audience and promotional efforts. Keep in mind that each marketplace has its own rules, regulations, and fees. You can also consider using a hosted e-commerce platform like Squarespace or Wix, which offer drag-and-drop website builders and integrated e-commerce features. These platforms are user-friendly and provide a range of templates and design options. However, you may find limitations in customization and integrations compared to self-hosted platforms. Lastly, consider selling through social media platforms like Instagram or Facebook. Both platforms offer shopping features that allow you to tag products in posts and stories, and customers can purchase from your feed directly. You can also use third-party apps to set up a shop on these platforms. Ultimately, the choice of e-commerce platform depends on your business needs, technical expertise, and target audience. You may choose to sell through a combination of platforms to maximize your reach. Which type of platform are you leaning towards, or would you like more information on a specific option?