Question

Consider the experiment of picking a four-digit PIN uniformly at random over all possible four-digit PINs, and define the random variables: X = number of distinct digits (i.e. how many different digits appear once or more) Y = length of longest streak of the same digit Below are the values of the random variables for some sample outcomes: X(1,3,1, 3)) = 2, Y((1,3,1,3)) = 1, X(2, 4, 3, 3)) = 3, Y(2, 4, 3, 3)) = 2, X (2,2, 4, 4)) = 2 Y(2, 2, 4, 4)) = 2 1. What is the set of possible values of X? A. {2,3} B. {2,3,4} C. {1,2,3,4} D. {0,1,2,3,4} 2. What is the value of the probability P({X = 1} {Y = 4})? A. 1/(9 x 8 x 7) B. 1/(10 x 9 x 8 x 7) C. 1/1,000 D. 1/10,000 3. Which of the following statements (if any) is true for any outcomes e S? A. X(s) > Y(s) B. X(s) <Y(s) + 2 C. X(s) +Y(s) > 3 D. None of the above

Answer 1

X a In can take 1, 2, 3, 4 There an be l'or 1 on 3 or 4 distinct numbers in a PIN - so, { 1, 2, 3, 4 } n I 2 That means The PIN is like 1,,1,13 $2,2,2, 23 Ete. In that cas there are such 10 s options And for all of them y=4 so, p 14x3 [Y=4]] - 10 - T 104 is the sample spree. loxlox loxlo for four spots 2 4 103

[so, A-2 Answer to ] 3.A 16 Then, X (s) = 1 Y68) = 4 So, No, it is false x(x) is not alway gretes. V (B) = 1 when, we have = { 2, 6, 8, 93 etc (ire all distines) Then, x (8)= 4 > Y(8) +2 So, 'B' is FALSE NAMIN

(9) If YCB) - 1 - Then, X (B) = 4, If (Yu) = 2) Then, X (8) can be 21 or 13 [ example Y (1) = 2 = {1, 1, 2,3}, {1,1, 2, 2} X(A) = 3 X(0)= 27 :. X (1) + V (A) = 4 or 5 STT

Y (1) 3 = X(0)=2 X (8) + V (0) = 5 Y (8) = 4 3 X (0) = 1 7) X (s) + Cs) = 5 so, Yes option & is True