Probability Theory and Statistics

  1. Your monthly electricity bill consists of a $20 service charge, plus $0.08 per kWh (kilowatt-hour) used. Suppose the number of kWh used per month is W = 800 + 10X , where X ∼ Binomial(40,0.5). Calculate: (a) E[C], the expected monthly cost of your electricity. (b) The standard deviation of C.
    (c) The probability P(C = E[C]).
  2. Imagine that, before a measurement of some (discrete) random variable X you are asked to guess the result. What should you choose? One strategy is to choose a number ν that is somehow “closest” to the most likely results of the measurement. Specifically, let the mean squared error be: ε=E[(X−ν)2]
    Explain the name and meaning of this quantity. Find the minimum of ε as a function of ν, by calculating the derivative dε/dν and setting it equal to zero. What is the significance of your result?
  3. Let X ∼ Geometric(1/2). Derive a (simple!) formula for the CDF of X, and plot your result.
  4. The CDF of the continuous random variable V is  0 v<−5 FV(v)= c(v+5)2 −5≤v<5

(a) Determine the value of the constant c required to make this CDF continuous. (b) What is P(V > 4)?
(c) What is fV (v)?
(d) CalculateE[V]andVar(V).

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5. The Rayleigh random variable X has a parameter a > 0, and PDF a2xea2x2/2 x≥0

fX(x)= 0 otherwise Prove that this PDF is normalized, and calculate the CDF of X.

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