1—Basic Stuff 291.40 If there are only 100 molecules of a gas bouncing around in a room, about how long will you have to wait to
find that all of them are in the left half of the room? Assume that you make a new observation every microsecond
and that the observations are independent of each other. Ans: A million times the age of the universe. [Care to
try 1023 molecules?]
1.41 If you flip 1000 coins 1000 times, about how many times will you get exactly 500 heads and 500 tails?
What if it’s 100 coins and 100 trials, getting 50 heads? Ans: 25, 8
1.42 (a) Use parametric differentiation to evaluate
∫
xdx. Start with∫
eαxdx. and then letα→ 0.
(b) Now that the problem has blown up in your face, change the integral from an indefinite to a definite integral
such as
∫b
a and do it again.1.43 The Gamma function satisfies the identity
Γ(x)Γ(1−x) =π/sinπxWhat does this tell you about the Gamma function of 1/2? What does it tell you about its behavior near the
negative integers? Compare this result to that of problem 16.
1.44 Start from the definition of a derivative, manipulate some terms, and derive the rule for differentiating the
functionh, whereh(x) =f(x)g(x)is the product of two other functions.
(b) Integrate the resulting equation with respect toxand derive the formula for integration by parts.