{"id":20300,"date":"2024-04-15T05:51:03","date_gmt":"2024-04-15T05:51:03","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20300"},"modified":"2024-04-15T05:51:03","modified_gmt":"2024-04-15T05:51:03","slug":"two-independent-random-variables-x-and-y-are-uniformly-distributed-in-the-interval-1-1-the-probability-that-max-x-y-is-less-than-frac12-is-a-frac34-b-frac916","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/two-independent-random-variables-x-and-y-are-uniformly-distributed-in-the-interval-1-1-the-probability-that-max-x-y-is-less-than-frac12-is-a-frac34-b-frac916\/","title":{"rendered":"Two independent random variables X and Y are uniformly distributed in the interval [-1, 1]. The probability that max [X, Y] is less than $$\\frac{1}{2}$$ is A. $$\\frac{3}{4}$$ B. $$\\frac{9}{{16}}$$ C. $$\\frac{1}{4}$$ D. $$\\frac{2}{3}$$"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;$$\\frac{3}{4}$$&#8221; option2=&#8221;$$\\frac{9}{{16}}$$&#8221; option3=&#8221;$$\\frac{1}{4}$$&#8221; option4=&#8221;$$\\frac{2}{3}$$&#8221; correct=&#8221;option1&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\frac{3}{4}}$.<\/p>\n<p>The probability that max [X, Y] is less than $\\frac{1}{2}$ is the same as the probability that both $X$ and $Y$ are less than $\\frac{1}{2}$. The probability that $X$ is less than $\\frac{1}{2}$ is $\\frac{1}{2}$, and the probability that $Y$ is less than $\\frac{1}{2}$ is also $\\frac{1}{2}$, since $X$ and $Y$ are independent. Therefore, the probability that max [X, Y] is less than $\\frac{1}{2}$ is $\\frac{1}{2} \\times \\frac{1}{2} = \\boxed{\\frac{1}{4}}$.<\/p>\n<p>Here is a more detailed explanation of each option:<\/p>\n<ul>\n<li>Option A: $\\frac{3}{4}$. This is the correct answer. The probability that max [X, Y] is less than $\\frac{1}{2}$ is the same as the probability that both $X$ and $Y$ are less than $\\frac{1}{2}$. The probability that $X$ is less than $\\frac{1}{2}$ is $\\frac{1}{2}$, and the probability that $Y$ is less than $\\frac{1}{2}$ is also $\\frac{1}{2}$, since $X$ and $Y$ are independent. Therefore, the probability that max [X, Y] is less than $\\frac{1}{2}$ is $\\frac{1}{2} \\times \\frac{1}{2} = \\boxed{\\frac{1}{4}}$.<\/li>\n<li>Option B: $\\frac{9}{{16}}$. This is not the correct answer. The probability that max [X, Y] is less than $\\frac{1}{2}$ is the same as the probability that both $X$ and $Y$ are less than $\\frac{1}{2}$. The probability that $X$ is less than $\\frac{1}{2}$ is $\\frac{1}{2}$, and the probability that $Y$ is less than $\\frac{1}{2}$ is also $\\frac{1}{2}$, since $X$ and $Y$ are independent. Therefore, the probability that max [X, Y] is less than $\\frac{1}{2}$ is $\\frac{1}{2} \\times \\frac{1}{2} = \\boxed{\\frac{1}{4}}$. This is not equal to $\\frac{9}{{16}}$.<\/li>\n<li>Option C: $\\frac{1}{4}$. This is not the correct answer. The probability that max [X, Y] is less than $\\frac{1}{2}$ is the same as the probability that both $X$ and $Y$ are less than $\\frac{1}{2}$. The probability that $X$ is less than $\\frac{1}{2}$ is $\\frac{1}{2}$, and the probability that $Y$ is less than $\\frac{1}{2}$ is also $\\frac{1}{2}$, since $X$ and $Y$ are independent. Therefore, the probability that max [X, Y] is less than $\\frac{1}{2}$ is $\\frac{1}{2} \\times \\frac{1}{2} = \\boxed{\\frac{1}{4}}$. This is not equal to $\\frac{1}{4}$.<\/li>\n<li>Option D: $\\frac{2}{3}$. This is not the correct answer. The probability that max [X, Y] is less than $\\frac{1}{2}$ is the same as the probability that both $X$ and $Y$ are less than $\\frac{1}{2}$. The probability that $X$ is less than $\\frac{1}{2}$ is $\\frac{1}{2}$, and the probability that $Y$ is less than $\\frac{1}{2}$ is also $\\frac{1}{2}$, since $X$ and $Y$ are independent. Therefore, the probability that max [X, Y] is less than $\\frac{1}{2}$ is $\\frac{1}{2} \\times \\frac{1}{2} = \\boxed{\\frac{1}{4}}$. This is not equal to $\\frac{2}{3}$.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;$$\\frac{3}{4}$$&#8221; option2=&#8221;$$\\frac{9}{{16}}$$&#8221; option3=&#8221;$$\\frac{1}{4}$$&#8221; option4=&#8221;$$\\frac{2}{3}$$&#8221; correct=&#8221;option1&#8243;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[691],"tags":[],"class_list":["post-20300","post","type-post","status-publish","format-standard","hentry","category-probability-and-statistics","no-featured-image-padding"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Two independent random variables X and Y are uniformly distributed in the interval [-1, 1]. The probability that max [X, Y] is less than $$\\frac{1}{2}$$ is A. $$\\frac{3}{4}$$ B. $$\\frac{9}{{16}}$$ C. $$\\frac{1}{4}$$ D. $$\\frac{2}{3}$$<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/two-independent-random-variables-x-and-y-are-uniformly-distributed-in-the-interval-1-1-the-probability-that-max-x-y-is-less-than-frac12-is-a-frac34-b-frac916\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Two independent random variables X and Y are uniformly distributed in the interval [-1, 1]. 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The probability that max [X, Y] is less than $$\\frac{1}{2}$$ is A. $$\\frac{3}{4}$$ B. $$\\frac{9}{{16}}$$ C. $$\\frac{1}{4}$$ D. $$\\frac{2}{3}$$","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/two-independent-random-variables-x-and-y-are-uniformly-distributed-in-the-interval-1-1-the-probability-that-max-x-y-is-less-than-frac12-is-a-frac34-b-frac916\/","og_locale":"en_US","og_type":"article","og_title":"Two independent random variables X and Y are uniformly distributed in the interval [-1, 1]. 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