Statistics questions

Question

1

A discrete random variable X models how many sales per day a salesman makes. From past experience we know PX(X ≤ 3) = 0.3 and PX(X ≥ 3) = 0.9. Which of the following must be true?

PX(X < 4) < 0.6

PX(X = 3) = 0.1

PX(X = 3) = 0.2

X must follow a Binomial Distribution.

2

Let Z ∼ N(−1, 2) and put X = 2(2 + Z). Which one of the following statements is true?

X ∼ N(−2, 4).

X ∼ N(2, 4).

X ∼ N(−2, 8)

X ∼ N(2, 8)

Answer 1

1) P(X<=3) + P(x>=3) = .3+.9 = 1.2. This is basically the probability of everything possible with the probability of 3 added in twice. The probability of everything possible is P(X<3) + P(x>=3) = 1. Since the probability of everything + P(X=3) = 1.2, the P(X=3) = 1.2 - 1 = 0.2. The answer is P(X=3) = 0.2

2) Z~N(-1,2) tells us that Z is a random variable that is approximately normally distributed with mean = -1 and standard deviation = 2. X is a linear transformation of Z. This means that x will have the same shape as Z - approximately normally distributed and the mean of X is the same linear transformation of the mean of Z: µ_X = 2(2 + -1) = 2. The standard deviation of a linear transformation is only affected by multiplying or dividing - measures of spread are not affected by adding or subtracting constants in a linear transformation. Thus σ_X = 2(2) = 4. Thus X is approximately normally distributed with mean = 2 and standard deviation = 4. The answer is X~N(2,4).