Random walks, Markov chains and martingales

37 problems

Gambler's ruin, waiting times for patterns, branching processes and Brownian motion. Most of these reduce to a recurrence, a stationary distribution or a martingale that stays constant on average. Each problem has hints and a worked solution, free, no account.

#titlelevel
7 Taking turns until head then tail medium
13 The area under a Brownian path medium
16 Always ahead in the count medium
24 A twelve or two sevens medium
30 Back at the start of a tetrahedron medium
42 The first total past 100 hard
43 Reaching one before time one medium
46 Two looks at one path easy
49 Positive at both times medium
64 Who keeps the ball medium
69 Visiting every hour hard
76 Waiting for HTHT medium
92 One step from the edge medium
93 Two sixes in a row medium
94 Repainting until all match medium
110 The last hour on the clock medium
114 A knight finds its way home hard
122 Chocolates and jelly beans easy
129 Coming back before the end medium
132 Variance of an Itô integral medium
143 Back at twelve after twelve steps medium
167 A walk on the keypad easy
181 The mean of exp of Brownian motion easy
185 The sum up to the first six easy
195 Two rolls two apart medium
196 Across the cube medium
204 Lost in a cave easy
208 The sixth moment of Brownian motion easy
236 Against the drift hard
238 Win by two easy
241 Will the amoebas die out? medium
247 Waiting for tail then head easy
254 Calling the next red card medium
259 How long the walk lasts easy
263 A race between two patterns hard
269 Confidence that feeds on itself medium
274 Ruin with an edge medium

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