Question

The random variables X1 and X2 both follow normal distributions. The mean of X1 is E(X1)=5,...

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
Math   Statistics-And-Probability   
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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

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