Question

Let ? be the number that shows up when you roll a fair, six-sided die, and, let ? = ?^2 − 5? + 6.

a. Find both formats for the distribution of ?. (Hint: tep forms of probability distributions are CDF and pmf/pdf.)

b. Find F(2.35)

Answer 1

If 'X' is the number that shows up when you roll a fair, six-sided die then the possible values of 'X' are -

X = 1,2,3,4,5,6.

All the six possibilities are equi-probable because the die is fare.

So, P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = P(X = 5) = P(X = 6) = 1/6.

The values of Y for each X is shown below -

For X = 1, Y = (1)² - 5(1) + 6 = 2

For X = 2, Y = (2)² - 5(2) + 6 = 0

For X = 3, Y = (3)² - 5(3) + 6 = 0

For X = 4, Y = (4)² - 5(4) + 6 = 2

For X = 5, Y = (5)² - 5(5) + 6 = 6

For X = 6, Y = (6)² - 5(6) + 6 = 12

Note that Y is taking same values for X = 1 and X = 4.

So, P(Y = 2) = P(X = 1) + P(X = 4) = 2/6 = 1/3

Similarly, P(Y = 0) = P(X = 2) + P(X = 3) = 2/6 =1/3

And, P(Y = 6) = P(Y = 12) = 1/6.

Hence, the PMF and CDF of the random variable Y can be written as -

PMF -

CDF -

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b)

From the CDF, we can see that -

F_Y(y) = 2/3 for 2 y < 6

Hence, F(2.35) = 2/3 = 0.6667.

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