Question

f(x) = (kx + 1)/25 , x = 0, 1, 2, 3, 4.
1. Find the value of k that makes f a probability mass function. Graph f.
2. What is the expected number of days a rat will spend in the maze?
3. Find and graph the cumulative distribution of the number of days a rat
will spend in the maze. Conditional on the fact that a rat spends at most
3 days in the maze, what is the probability that a rat spends two days in?
4. Suppose that a rat is to receive from this researcher a food that costs 10
dollars for each of the first two days spent in the maze and 5 dollars for
each day after the first two days, what is the expected expenditure of this
researcher?

Answer 1

golution - 1) Wikiy EPCT) = 1 E (x+1) 25 R=0,1,2,3,4 K(1) +1 + k(3)41 KCO)+1 25 + + + K(Q)+1 25 K(4) +1 25 25 25 JOK+G 25 10k +5 = 25 10K - 20 ) f(a) 92+1 12=0,1,2,3,4 25 o) Expected no. of dogs ECX) = Ex fca) & 10)+1 ° ( 869') + 1 (15)+2 (4) 13 () +4 () 0+ 3 25 36 + 10 80 + ง 25 + 70 2.8 c) pca) = KX+ 25 2x+1 25 2:011121314 fco) eet 95 f(3) 25 f(i) = 3 25 f(2)= 25 fcm): 25

Cumulative distribution from is 25 fco) = fco) = 0.04 F(t) = fro)+ fci) - ja to to :0-16 FCQ) : fco+ fi+fca) = 9 95 0.36. f(3) = 400++1)+ f()+ f(3) = 16 :069 25 FC4) = f(o) + f(0)+(8) +-403) + f(4) 95 0.4 0.8 of 0.67 06 of 0.9 0.2 0.1 3 ga $ CX= 2 1433) PCX=D X 53) PCX53) PC) PY 3 3) fcx=) 3 Fox=3) 5125 0.64 20.3125

d Eapected Expenditure 3) 14 10 x fco)+ (10+10)$60) + (10+10+5) f(a) f C10+10 + 5+5) f(3) + (10 +10+5+5+5) fcu) = 10 (85) + 20 (35) + 25 ( 35) + 30636) + 35 (25) f 790 19 988 25