Question

The random variables X₁ and X₂ both follow normal distributions. The mean of X₁ is E(X₁)=5, and its variance is V(X₁)=2 The mean of X₂ is E(X₂)=9, and its variance is V(X₂)=3. If Y is a random variable such that Y = 3X₁+5X₂, what is P(Y<70)?

A student takes 4 measurements and finds that the mean is 64 and the sample variance is 81. What is the sample standard deviation

For a random variable X, which statement is most likely to be true

		If you observe X multiple times the value you get will always equal the mean
		If you observe X multiple times you will get the same value
		If you observe X multiple times you will get different values
		If you observe X multiple times the value you get will always equal the standard deviation

Answer 1

Since X₁ and X₂ both follow normal distributions, then Y = 3X₁+5X₂  (linear combination of normal random variables) follow normal distribution.

E[Y] = E[ 3X₁+5X₂] = 3E[X1] + 5E[X2]

= 3 * 5 + 5 * 9 = 60

V(Y) = V[ 3X₁+5X₂] = 3²V[X1] + 5²V[X2] = 9V[X1] + 25V[X2] = 9 * 2 + 25 * 3 = 93

Standard deviation of Y is = 9.64

P(Y < 70) = P[Z < (70 - 60)/9.64]

= P[Z < 1.037]

= 0.8502

Sample standard deviation = = 9

Sample standard deviation of mean = = 4.5

For a random variable X,  If you observe X multiple times you will get different values