Example 26 - Let X denote number of hours you study during - Examples

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Example 26 Let X denote the number of hours you study during a randomly selected school day. The probability that X can take the values x, has the following form, where k is some unknown constant. P(X = x) = 0.1 , ๐‘–๐‘“ ๐‘ฅ=0๏ทฎ๐‘˜๐‘ฅ, ๐‘–๐‘“ ๐‘ฅ=1 ๐‘œ๐‘Ÿ 2๏ทฎ๐‘˜ 5โˆ’๐‘ฅ๏ทฏ, ๐‘–๐‘“ ๐‘ฅ=3 ๐‘œ๐‘Ÿ 4๏ทฎ0, ๐‘œ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘ค๐‘–๐‘ ๐‘’๏ทฏ๏ทฏ (a) Find the value of k Making in tabular format Since X is a random variable , its Sum of Probabilities is equal to 1 0๏ทฎ4๏ทฎ๐‘ƒ(๐‘‹)๏ทฏ = 1 P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 1 0.1 + k + 2k + 2k + k = 1 6k = 1 โ€“ 0.1 6k = 0.9 k = 0. 9๏ทฎ6๏ทฏ k = 0.15 Example 26 (b) What is the probability that you study atleast two hours? Exactly two hours? At most 2 Hours Our probability distribution table is P(you study atleast two hours) = P(X โ‰ฅ 2) = P(X = 2) + P(X = 3) + P(X = 4) = 2k + 2k + k = 5k = 5 ร— 0.15 = 0.75 P(you study exactly two hours) = P(X = 2) = 2k = 2 ร— 0.15 = 0.30 P(you study atmost two hours) = P(X โ‰ค 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.1 + k + 2k = 0.1 + 3k = 0.1 + 3 ร— 0.15 = 0.1 + 0.45 = 0.55

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