Test 1 sample

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Stats 2D03 sample test 1 October 2023 There are four problems on this test, worth 10, 10, 15, and 15 points, respectively. If you want to have a timed practice, try completing this sample midterm in an hour and 15 minutes. 1. (3+4+3 points) Consider the function defined by f ( x ) = 1 / 2 0 x < 1 2 C 1 2 x 1 0 otherwise. (a) Find the constant C that makes f ( x ) the probability density function of a continuous random variable. (b) Suppose X is a continuous random variable with probability density function f ( x ), using the constant C found in part (a). Write down the cumulative distribution function of X . (c) The median of this probability distribution is the value m such that P { X m } = 50% = P { X > m } . Find the median of this probability distribution. 2. (2+4+4 pts) You have a bag containing three marbles numbered 0, 1, and 2. You sample 5 times from the bag, with replacement. (a) How many outcomes are in my sample space for this experiment? (b) What is the probability that you never pick the same marble twice in a row? (c) What is the probability that the numbers that you picked add up to 5? 3. (3+3+4 pts) We roll a 4-sided die 5 times. Let X denote the number of '1's rolled and let Y denote the number of '2's. (a) Compute P { X = 3 } . (b) Compute P ( { X = 0 } ∪ { Y = 0 } ) . (c) Are X and Y are independent random variables? Prove or disprove your answer. 4. (3+3+4 pts) Aragorn and Bilbo have two coins: one is fair and the other is biased to give 'H' with 80% probability. Bilbo chooses between the two coins uniformly at random and Aragorn takes the other coin. They each flip their coin once. (a) Calculate the probability that Bilbo flips 'H'. (b) Find the conditional probability that Bilbo chose the fair coin, given that he got 'H'. (c) Find the conditional probability that Aragorn gets 'H' given that Bilbo got 'H'.
Stats 2D03 Test 1 formula sheet Noah Forman Discrete Uniform - U Uniform( { 1 , 2 , . . . , n } ) PMF: p U ( k ) = 1 n for k ∈ { 1 , . . . , n } , E [ U ] = n + 1 2 . Bernoulli - I Bernoulli( p ) PMF: p I ( k ) = 1 p if k = 0 , p if k = 1 , for k ∈ { 0 , 1 } , E [ I ] = p. Binomial - S Binomial( n, p ) PMF: p S ( k ) = n k p k (1 p ) n k for k ∈ { 0 , 1 , . . . , n } , E [ S ] = np. Geometric - G Geometric( p ) PMF: p G ( k ) = (1 p ) k 1 p for k ∈ { 1 , 2 , 3 , . . . } , E [ G ] = 1 p . Continuous Uniform - U Uniform[ a, b ] PDF: f U ( x ) = 1 / ( b a ) if x [ a, b ] , 0 otherwise. CDF: F U ( x ) = 0 if x < a, x a b a if x [ a, b ] , 1 if x > b. E [ U ] = a + b 2 . ( x + y ) n = n X k =0 n k x k y n k . P n [ j =1 A j ! = X J ⊆{ 1 ,...,n } , J ̸ = ( 1) | J | +1 P \ i J A i ! = P ( A 1 ) + P ( A 2 ) + · · · + P ( A n ) ( n terms ) P ( A 1 A 2 ) − · · · − P ( A n 1 A n ) n 2 terms + P ( A 1 A 2 A 3 ) + · · · + P ( A n 2 A n 1 A n ) n 3 terms . . . + ( 1) n +1 P ( A 1 A 2 ∩ · · · ∩ A n ) .
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