Question

3.1 There is a random variable X with observations {X1,X2, ..., Xn). It is known that these observations follow the normal distribution with mean μ and variance σ2. Which of the following will lead to a standard normal distribution? (a) (X-A)/o (b) (X- )/a2 (c) (X + μ)/o2 (d) (X + μ)/σ 3.2 In standard normal distribution, 99.7% of observations lie in the range between 3.3 A cumulative distribution function of a random variable Xis by definition a probability that X will take a value (a) less than x (b) less than or equal to x c) x (d) greater than r (e) greater than or equal to r

Answer 1

Answer

3.1

In order to convert normal distribution to a standard normal distribution, we have to form new normal distribution such that mean gets equal to 0 and Variance = 1. It is done using a Following Formula:

Z = (X - u)/ $\sigma$ . Here Z will follow standard normal distribution

E(Z) = (1/ $\sigma$ )(E(X) - E(u)) = (1/ $\sigma$ )(u - u) = 0

Var(Z) = Var((X - u)/ $\sigma$ ) = (1/ $\sigma ^2$ )Var(x) = $\sigma ^2$ / $\sigma ^2$ = 1

Formula :

Var(aX - b) = a²Var(X)

E(X + Y) = E(X) + E(Y)

3.2)

We have to find a such that P(-a < X < a) = 0.997

P(-a < X < a) = P(X < a) - P(X < -a) = 0.997. We can see from standard normal table that

P(X < 3) = 0.998 and P(X < -3) = 0.001

=> P(-3 < X < 3) = P(X < 3) - P(X < -3) = 0.997

Hence the correct answer is (c) [-3 , 3]

3.3)

Cumulative distribution function is defined as a probability that X will take a value less than or equal to x.

Hence, the correct answer is (b) less than or equal to x