Question

Problems 27: the number X is the number of car which passed through the

Answer 1

Answer:

a) The value of a= 0.59

b)  The probability that there are at-least 3 cars passing through the stop sign

P(x>3)  = 0.03

c)

The Expected value of X  = 0.62

d)

 The variance of X is   σ² = 0.9556

e)

The standard deviation of X

                                    σ = 0.9775

Step-by-step explanation:

Given data

X       :      0         1          2            3        4          5

P(x)   :    a       0.30     0.05    0.03    0.02     0.01

a)

      ∑ $p(x_{i} )$ = 1

 a + 0.30+ 0.05+ 0.03+ 0.02+0.01 = 1

 a  + 0.41 = 1

     a  = 1 - 0.41

     a  = 0.59

b)

 The probability that there are at-least 3 cars passing through the stop sign

P(x >3)  = P( x=4) + P( x=5)

           =   0.02 +0.01

          = 0.03

c)

X       :      0           1          2            3        4          5

P(x)   :      0.59      0.30     0.05    0.03    0.02     0.01

The Expected value of X

E(X) = ∑ x P(X= x)

      =  0 + 1 ×0.30 + 2×0.05 + 3×0.03 + 4×0.02 + 5×0.01

     =   0.30 + 0.1 + 0.09 +0.08 +0.05

    =  0.62

The Expected value of X

                          E(X)  = 0.62

d)

   The variance of the discrete distribution

       σ² =   ∑ x²p(x) -μ²

      σ² =   0 + 1² ×0.30 + 2² ×0.05 + 3² ×0.03 + 4² ×0.02 + 5²× 0.01 - (0.62)²

          =   1.34 - 0.3844

         = 0.9556

      σ² = √0.9556

e) The standard deviation of the  discrete distribution

     σ = 0.9775