Question

(a) Let Y be distributed N(0, 1), i.e. the standard normal variable. Calculate P(Y <= 1.96), P(Y <= 0) and P(Y > 0.6772). Bound P(Y > 3).

(b) Let Y be distributed N(1, 4). Calculate P(Y < 1), P(Y > 2), P(1 ≤ Y ≤ 2).

(c) Let Y be distributed χ 2 1 . Calculate P(Y < 1), P(Y > 3.8416) using the results of the first question.

(d) Let Y be distributed χ 2 3 . Calculate P(Y < 7.81), P(Y > 11.34).

Answer 1

Z TABLE:

(a) GIVEN:

  Let Y be distributed N(0, 1). Thus Y follows standard normal distribution with mean =​ 0 and variance ​=1

• ​

{Since Z=(Y-mu)/sigma}

​

   {Using Z table with row value 1.9 and column corresponding to 0.06}

• 

  

{Using Z table with row value 0.5 and column corresponding to 0.00}

• 

  {Since Z=(Y-mu)/sigma}

   {Using Z table with row value 0.6 and column corresponding to 0.08}

• 

     {Since Z=(Y-mu)/sigma}

  

  

{Using Z table with row value 3.0 and column corresponding to 0.00}​

  

(b) GIVEN:

  Let Y be distributed N(1, 4).​ Thus Y follows normal distribution with mean=​ 1 and variance ​=4. Thus =2

• 

  {Since Z=(Y-mu)/sigma}

{Using Z table with row value 0.0 and column corresponding to 0.00}​

• 

  {Since Z=(Y-mu)/sigma}

   {Using Z table with row value 0.5 and column corresponding to 0.00}​

  

  

• 

   {Since Z=(Y-mu)/sigma}

  

  

  

  

  

(c) GIVEN:

Let Y follows ​ with 1 degree of freedom.

• 

{Let Y=Z^2 as square of standard normal distribution gives chi-square distribution. Here Y~ ​ distribution with 1 degree of freedom and Z~N(0,1)}

  

  

{Using Z table value with row corresponding to 1.0 and column corresponding to 0.00}

•   

  

  

  

  

{Using Z table value with row corresponding to 1.9 and column corresponding to 0.06}

  

(d) GIVEN:

Y follows distribution with 3 degrees of freedom. Then its normal approximation is given by

​ where n is degrees of freedom of distribution.

• 

{since n=3}

  

  

{Using Z table value with row corresponding to 1.9 and column corresponding to 0.06}

  

• 

  

  

  

{Using Z table value with row corresponding to 3.4 and column corresponding to 0.00}