Three Statistics questions
1. An electronics company manufactures 3 models of microprocessors. The models sell for
the following prices: Model A: $20, Model B: $40 and Model C: $60.
Customers buying microprocessors from this company choose the various models with the following probabilities:
P(A)=0.2 P(B)=0.7 P(C)=0.1
(a) The plot below shows the probability density function f(x) for the maximum sus- tainable clock speed of the model A microprocessor.
(b) Find the mean of the maximum sustainable clock speed for the model A micropro- cessor.
(c) Find the cumulative distribution function F (x) for this random variable.
(d) Compute the probability that the maximum sustainable clock speed for a model Awill be less than 60 kHz.
(e) Compute the probability that the maximum sustainable clock speed for a model A will be between 60 and 80 kHz.
2. A bag contains 7 red and 3 blue marbles. You draw three marbles from this bag without replacement. Let X be the number of blue marbles you get.
(a) Is X a discrete or a continuous random variable?
(b) Determine the probability distribution function f(x) and the cumulative distribu-tion function F(x) for the random variable X.
(c) Using F(x) find P(X ≤ 1)
(d) Compute the mean, variance and standard deviation of the random variable X.
(e) Assume that this is a game in a casino: the payoff is $100X and it costs $10, 000 to play the game each time. Compute the average net gain/loss per game.
3. A cellphone service provider wants to analyze the signal strengths of its network.
(a) Let X be the random variable that is the distance (in miles) that a customer will be to the nearest cellphone tower. Analyzing their database, the company finds that the probability density function is
f(x) = 1/4, 2≤x≤6
f(x) = 0, otherwise
Find the mean and variance of random variable X.
(b) The company is now interested in signal strengths. The company finds that the signal power (in milliwatts) at a distance x from the tower is P(x) = 40/(x^2). Given the probability density function for X in the previous part, what is the expected signal strength? Hint: This is asking for E[P(x)]
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