# Normal Distribution

• Apr 2nd 2011, 06:07 AM
teddybear67
Normal Distribution
Would really appreciate if someone could help !!!

1) If a large number of samples of size n are taken from B(20,0.2) and approximately 90% of the sample means are less than 4.354, estimate n.

2) i. The random variable $\displaystyle X$ has the distribution $\displaystyle N(50, 8^2)$. Given that $\displaystyle X_{1}$ and $\displaystyle X_{2}$ are to independent observations of $\displaystyle X$ , find $\displaystyle P( X_{1} > X_{2} + 15)$
ii. The random variable $\displaystyle Y$ is related to $\displaystyle X$ by the formula $\displaystyle Y = aX + b$, where $\displaystyle a$ and $\displaystyle b$ are constants with $\displaystyle a > 0$. Given that $\displaystyle P(Y < 74) = P(Y > 146) = 0.00668$, find the values of $\displaystyle E(Y)$ and $\displaystyle Var(Y)$ , and hence find the values of a and b.

thanks!!
• Apr 3rd 2011, 05:34 AM
Sambit
1) If X is the corresponding random variable, then $\displaystyle P(\bar{X}<4.354)=0.90$

=> $\displaystyle P(\frac{\bar{X}-20}{\sqrt{\frac{0.2}{n}}}<\frac{4.354-20}{\sqrt{\frac{0.2}{n}}})=0.90$

=> $\displaystyle \Phi(\frac{4.354-20}{\sqrt\frac{0.2}{n}})=0.90$

Find the inverse cdf of a standard normal distribution having cumulative probability $\displaystyle 0.90$; that will be equal to $\displaystyle \frac{4.354-20}{\sqrt\frac{0.2}{n}}$. Then solve for $\displaystyle n$.

2)i) HINT: You have to find $\displaystyle P(X_1-X_2>15)$ where $\displaystyle (X_1-X_2)\sim N(50-50 , 8^2+8^2)$

ii) HINT: The distribution of Y is symmetric about $\displaystyle \frac{74+146}{2}$. So the mean of $\displaystyle Y$ is $\displaystyle \frac{74+146}{2}$. For variance, replace $\displaystyle Y$ by $\displaystyle X$ and proceed.
• Apr 4th 2011, 05:11 AM
teddybear67
for 1), i did as you taught me to, but the answer given for n is 42 whereas i got 0.00134182 oO

and as for the rest, i got them! thanks!!
• Apr 4th 2011, 05:31 AM
Sambit
Horrible mistake! I read it normal instead of binomial! The LHS will be

$\displaystyle \Phi(\frac{4.354-(20*0.2)}{\sqrt{\frac{20*0.2*(1-0.2)}{n}}})$
• Apr 4th 2011, 08:48 PM
teddybear67
Ok! I got it!! Thanks!!