Probability distribution of a random variable

Econ2300
assignment: Ch5 Quiz

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Exercise 5.12 METHODS AND APPLICATIONS
Suppose that the probability distribution of a random variable x can be described by the formula

P(x) = x
________________________________________
15

for each of the values x = 1, 2, 3, 4, and 5. For example, then, P(x = 2) = p(2) =2/15.

(a) Write out the probability distribution of x. (Write all fractions in reduced form.)

x 1   2   3   4   5

P(x) ________________________________________   ________________________________________   ________________________________________   ________________________________________   ________________________________________

________________________________________

(b) Show that the probability distribution of x satisfies the properties of a discrete probability distribution.(Round other answers to the nearest whole number. Leave no cells blank – be certain to enter “0” wherever required.)

P(x) ≥   for each value of x.

 

(c) Calculate the mean of x. (Round your answer to 3 decimal places.)

µx

(d) Calculate the variance, σ2x , and the standard deviation, σx. (Round your answer σx2 in to 3 decimal places and round answer σx in to 4 decimal places.)

σx2

σx

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Exercise 5.23 METHODS AND APPLICATIONS
Suppose that x is a binomial random variable with n = 5, p = 0.3, and q = 0.7.

(b)  For each value of x, calculate p(x), and graph the binomial distribution. (Round final answers to 5 decimal places.)

p(0) =  , p(1) =  , p(2) =  , p (3) =  ,
p(4) =  , p(5) =

(c) Find P(x = 3). (Round final answer to 5 decimal places.)

P(x=3)

(d) Find P(x ≤ 3). (Do not round intermediate calculations.  Round final answer to 5 decimal places.)

P(x ≤ 3)

(e) Find P(x < 3). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x < 3) = P(x ≤ 2)

(f) Find P(x ≥ 4). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x ≥ 4)

(g) Find P(x > 2). (Do not round intermediate calculations. Round final answer to 5 decimal places.)

P(x > 2)

(h) Use the probabilities you computed in part b to calculate the mean, μx, the variance, σ 2x, and the standard deviation, σx, of this binomial distribution. Show that the formulas for μx , σ 2x, and σx given in this section give the same results. (Do not round intermediate calculations. Round final answers to µx and σ 2x in to 2 decimal places, and σx in to 6 decimal places.)

µx

σ2x

σx

(i) Calculate the interval [μx ± 2σx]. Use the probabilities of part b to find the probability that x will be in this interval. Hint: When calculating probability, round up the lower interval to next whole number and round down the upper interval to previous whole number. (Round your answers to 5 decimal places. A negative sign should be used instead of parentheses.)

The interval is [ ,  ].

P(  ≤ x ≤  ) =

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MC Qu. 14 The mean of the binomial distribution is equ…
The mean of the binomial distribution is equal to:

p

np

(n) (p) (1-p)

px (1-p)n-x
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