Determine whether the random variable is discrete D or Continuous C (a) The numb

Question

Determine whether the random variable is discrete D or Continuous C (a) The number of days with measurable rainfall in Honolulu, Hawaii during a year ____ (b) The miles per gallon of gasoline obtained by a randomly selected Toyota Prius ____ (c) The number of golf balls hit into the ocean on the 18th hole at Pebble Beach this month ____. d) The weight (in grams) of a randomly selected robin's egg ____ At the Wimbledon Tennis Championship to win a match in nens singles a player must win the best of five following data represent the number of sets played x in the mens singles final match for the years 1968 to 2011: (a) Construct a probability model for the random variable x the number of sets played in the Wimbledon mens singles final match. (b) Draw a probability histogram. For data of problem 1) (a) Compute the mean of the random variable x. (b) Compute the standard deviation of the random variable x.

Explanation / Answer

1. Discrete variable: Frequencies are defined only for certain values of x.in the given interval.

Continuous variable: Frequencies are defined for all values of x in the given interval.

(a) number of days is discrete. REASON: x is restricted to integers.

(b) miles per gallon is continuous for positive values of x. REASON: miles can be expressed as any positive decimal number.

(c) number of golf balls is discrete. REASON: x is restricted to integers.

(d) weight is continuous for positive value of x. REASON: weight can be expressed as any positive decimal number.

2.

    x                    frequency

   3                      19

   4                        12

   5                       13

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Total frequncy = 44.

To get the pdf, divide each frequency by total frequency

x           Frequency              f(x)

3                 19                  19/44 = 0.4318

4                 12                  12/44 = 0.2727

5                  13                  13/44 = 0.2955

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Total                                      = 1.0000

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(b) Guidelines for drawing probability histogram:
First draw x and y axes - posive sides only - intersecting on LHS bottom at the origin.

On the x axis, mark 5 points at equal distances: 1,2,3,4,5

On the y axis, mark the points: 0.1, 0.2, 0.3, 04, 0.5 at equal distances.

Now:

Mark on the graph the first frequency as a point as given below:

x value 3 & y value 0.4318

Similarly, other 2 points on the graph paper.

Draw vertical lines at X = 3, x =4 and X = 5

Make them as 3 vertcal rectangles by joining the ends of the 3 frequenciies.

(3)

(a)

For getting mean = E(x):

x           f(x)       xf(x)              x2 f(x)

3          0.4318      1.2954             3.8862

4           0.2727      1.0908               4.3632

5           0.2955       1.4775              7.3875

Total                   = 3.8337               15.6369

So, mean = E(x) = 3.8337

To find SD:

E(X2) = 15.6369

s2 = E(x2) - (E(x))2= 15.6369 - (3.8337)2 = 15.6369 - 14.6973 = 0.9396

So,

sd = 0.9693

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