# probability-theory-and-statistics-6

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 âˆ¼ 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]).
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 Î½ that is somehow â€œclosestâ€ to the most likely results of the measurement. Specifically, let the mean squared error be: Îµ=E[(Xâˆ’Î½)2] Explain the name and meaning of this quantity. Find the minimum of Îµ as a function of Î½, by calculating the derivative dÎµ/dÎ½ and setting it equal to zero. What is the significance of your result?
Let X âˆ¼ Geometric(1/2). Derive a (simple!) formula for the CDF of X, and plot your result.
The CDF of the continuous random variable V is ï£±ï£² 0 v<âˆ’5 FV(v)= c(v+5)2 âˆ’5â‰¤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).
5. The Rayleigh random variable X has a parameter a > 0, and PDF a2xeâˆ’a2x2/2 xâ‰¥0
fX(x)= 0 otherwise Prove that this PDF is normalized, and calculate the CDF of X.

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