Question

Number of Blemishes (X) Probability f(x) 0 1 2 3 4 5 0.23 0.31 0.15 0.17 0.05 0.09 a) Show that the function $f(x)$ satisfies the properties of a probability mass function. [10 points] b) Determine the cumulative distribution function of the random variable $X$. [10 points] c) Find the mean (expected value), the variance, and the standard deviation of $X$. [15 points] d) Calculate the following probabilities: [15 points] 1. $P(X > 3)$ 2. $P(X = 6)$ 3. $P(X \ge 3 \cup X < 2)$ 4. $P(X > 1 \cap X < 4)$ 5. $P(1.5 \le X \le 3)$

          Number of Blemishes
(X)
Probability f(x)
0
1
2
3
4
5
0.23
0.31
0.15
0.17
0.05
0.09
a) Show that the function $f(x)$ satisfies the properties of a probability mass function. [10
points]
b) Determine the cumulative distribution function of the random variable $X$. [10 points]
c) Find the mean (expected value), the variance, and the standard deviation of $X$. [15 points]
d) Calculate the following probabilities: [15 points]
1.
$P(X > 3)$
2.
$P(X = 6)$
3.
$P(X \ge 3 \cup X < 2)$
4.
$P(X > 1 \cap X < 4)$
5.
$P(1.5 \le X \le 3)$

Number of Blemishes
(X)
Probability f(x)
0
1
2
3
4
5
0.23
0.31
0.15
0.17
0.05
0.09
a) Show that the function f(x) satisfies the properties of a probability mass function. [10
points]
b) Determine the cumulative distribution function of the random variable X. [10 points]
c) Find the mean (expected value), the variance, and the standard deviation of X. [15 points]
d) Calculate the following probabilities: [15 points]
1.
P(X > 3)
2.
P(X = 6)
3.
P(X ≥ 3 ∪ X < 2)
4.
P(X > 1 ∩ X < 4)
5.
P(1.5 ≤ X ≤ 3)

Added by John A.

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