# Poisson's Distribution and Normal Distribution

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ENGINEERING MATHEMATICS Poisson's Distribution and Normal Distribution

### Transcript of Poisson's Distribution and Normal Distribution

PowerPoint PresentationRAKESH TALREJA

Achievements:

Area of Expertise: Control Systems, Engineering Mathematics

Signals & Systems, Digital Electronics, EMT and General Aptitude

ABOUT ME

Poisson's Distribution and Normal Distribution

1. If X has a Poisson distribution with parameter λ and if P(X = 0) = 0.2, then P(X > 2) = ______.

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson Random Variables

Consider the case of Binomial RV. If ‘n’ becomes very

large, and ‘p’ is very small, the Binomial distribution

converges to Poisson distribution, given by

k = e−λλk

k! , k = 0, 1, 2,−− −n

Where, λ = np is the mean = 1 − ≅ =

Note: In practice, if ‘p’ is of order 0.01 or 0.001, and np ≤ 5, Poisson is used.

Poisson Random Variables

Poisson's Distribution and Normal Distribution

2. At a busy traffic intersection, the probability p of an individual car having an accident is p = 0.0001. If 1000 cars pass through the intersection between 5 PM and 6 PM every day, then the probability that two or more car accidents occur during that period is _____. (Use Poisson distribution).

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

3. What is the probability that at most 5 defective fuses will be found in a box of 200 fuses, if 2% of such fuses are defective?

A. 0.82

B. 0.79

C. 0.59

D. 0.52

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

4. A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is ________.

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

5. The number of accidents occurring in a plant in a month follows Poisson distribution with mean as 5.2. The probability of occurrence of less than 2 accidents in the plant during a randomly selected month is

A. 0.029

B. 0.034

C. 0.039

D. 0.044

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

6. On an average two covid positive cases are received per day in a hospital. The ratio of the probability of two or less covid positive cases in 2 consecutive days to the probability of exactly two cases on a given day is _______

A. 6.5e–4

B. 6.5e–2

C. 13e–2

D. 13e–4

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

7. If a random variable X has a Poisson distribution with mean 5, then the expectation E[(X + 2)2] equals _______ .

[GATE-2017, 2 MARKS]

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

8. If X is a Poisson’s random variable such that,

P(X = 2) = P(X = 4) + 15 P(X = 6)

Then, standard deviation of X is _______

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

9. If the second moment about origin is 6. The mean of a Poisson's Distribution is ________

A. –3 & 2

B. 2 only

C. –3 only

D. 3 & 2

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

10. Out of 1000 balls 50 are red and rest are white. If 80 balls are picked up at random, the probability of picking up not more than 3 red balls is ______

A.

B.

A normal RV has the PDF of the form

= 1

22 ,

2 is the variance, Var(X)

PDF: x = 1

2σ2 , for all x

P(X ≤ x) = ∞– x 1

2σ2 dx ⇒ can’t be directly integrated

A normal RV ‘’Z’’ is a standard normal RV if μ = 0, σ2 = 1

P.D.F. () =

–

2π e–z

The value of (z) are evaluated by numerical methods and

recorded in tables/calculators.

Var(Y) = a2Var (X)

Key Points

• Area (x ≥ μ) = Area (x ≤ μ) = 1/2

• P x ≥ μ = P x ≤ μ = 1/2

• ∞− μ

1

2

• If X is Normal R.V. having mean = (μ) and variance (σ2). Then

Y = a X + b is also a normal R.V. having the mean = (aμ + b) and

variance = (a2σ2)

Poisson's Distribution and Normal Distribution

• If X is the Normal R.V. having mean = (μ) and variance = (σ2) then

= –

is a standard normal R.V. having mean = 0 and variance = 1

• If X1 and X2 are two independent normal R.V. Then

= 11 + 22 is also a normal R.V. having

mean = a1μ1 + a2μ2 and variance = a1 2 σ1

2 + a2 2 σ2

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Key Points Approximately 68% of the values of any normal distribution

are located within one standard deviation about the mean.

P(μ – σ ≤ X ≤ μ + σ) = 0.6826 ≈ 68.26%

Approximately 95% of the values of any normal distribution

are located within two standard deviation about mean.

P(μ – 2σ ≤ X < μ + 2σ) = 0.9544 ≈ 95.44%

Approximately 99.7% of the value of any normal distribution

are located with in three standard deviation about the mean.

P(μ – 3σ ≤ X ≤ μ + 3σ) = 0.997 ≈ 99.7%

Poisson's Distribution and Normal Distribution

11. A random variable X has normal distribution with mean 100. If P(100 < X < 120) = a then P(X < 80) =

A. 1 – 2a

B. 1 – a

C. 1−2α

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

12. A normal random variable X has the following probability density function

A. 0

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

13. If X is a normal variate with mean 30 and standard deviation 5,

What is the probability P(26 ≤ X ≤ 34) ?

Given A(z > 0.8) = 0.2118.

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

14. A continuous random variable X is normally distributed with mean value 6 and standard deviation 2. Consider the following probabilities, R = P(4 < x < 10) & S = P(0 < X < 8)

Then which of the following is correct

A. R > S

B. R < S

[MSQ]

15. If the masses of 300 students are normally distributed with mean 68 kgs and standard deviation 3 kgs.

(Area under normal curve between z = 0 & z = 1.33 is 0.4082)

A. The number of students whose weight is greater than 72 kg is expected to be 28

B. The number of students whose weight is less than or equal to 64 kg is expected to be 28

C. The number of students whose weight lies between 65 & 71 kg is expected to be 205

D. The number of students whose weight lies between 62 & 74 kg is expected to be 286

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

16. Let X1, X2 be two independent normal random variables with means μ1, μ2 and standard deviations σ1, σ2, respectively. Consider Y = X1 – X2; μ1 = μ2 = 1, σ1 = 1, σ2 = 2. Then,

A. Y is normally distributed with mean 0 and variance 1

B. Y is normally distributed with mean 0 and variance 5

C. Y has mean 0 and variance 5, but is NOT normally distributed

D. Y has mean 0 and variance 1, but is NOT normally distributed

[GATE-2018, 1 MARK]

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

17. Let X be a random variable following normal distribution with mean + 1 and variance 4. Let Y be another normal variable with mean -1 and variance unknown.

If ( ≤ −1) = ( ≥ 2), the standard deviation of Y is

A. 3

B. 2

C. √2

D. 1

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

18. If f(x) = k exp. {–(9x2 – 12x + 13)}, is a p.d.f. of a normal distribution (k, being a constant), the mean and standard deviation of the distribution:

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

19. The lengths of a large stock of titanium rods follow a normal distribution with a mean (μ) of 440 mm and a standard deviation (σ) of 1mm. What is the percentage of rods whose lengths lie between 438 mm and 441 mm?

A. 81.85%

B. 68.4%

C. 99.75%

D. 86.64%

Binomial Distribution Approximation

If be a Binomial R.V. with parameters n & p,

Then for Large values of n, can be approximated as

Normal R.V.

Poisson's Distribution and Normal Distribution

20. Assume that 4 percent of the population over 50 years old has glasses. If a random sample of 3500 people over 50 years is taken, then the probability that less than 150 of them have glasses is _______. (Area under the normal curve between in Z = 0 and Z= 0.86 is 0.3051).

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Achievements:

Area of Expertise: Control Systems, Engineering Mathematics

Signals & Systems, Digital Electronics, EMT and General Aptitude

ABOUT ME

Poisson's Distribution and Normal Distribution

1. If X has a Poisson distribution with parameter λ and if P(X = 0) = 0.2, then P(X > 2) = ______.

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson Random Variables

Consider the case of Binomial RV. If ‘n’ becomes very

large, and ‘p’ is very small, the Binomial distribution

converges to Poisson distribution, given by

k = e−λλk

k! , k = 0, 1, 2,−− −n

Where, λ = np is the mean = 1 − ≅ =

Note: In practice, if ‘p’ is of order 0.01 or 0.001, and np ≤ 5, Poisson is used.

Poisson Random Variables

Poisson's Distribution and Normal Distribution

2. At a busy traffic intersection, the probability p of an individual car having an accident is p = 0.0001. If 1000 cars pass through the intersection between 5 PM and 6 PM every day, then the probability that two or more car accidents occur during that period is _____. (Use Poisson distribution).

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

3. What is the probability that at most 5 defective fuses will be found in a box of 200 fuses, if 2% of such fuses are defective?

A. 0.82

B. 0.79

C. 0.59

D. 0.52

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

4. A traffic office imposes on an average 5 number of penalties daily on traffic violators. Assume that the number of penalties on different days is independent and follows a Poisson distribution. The probability that there will be less than 4 penalties in a day is ________.

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

5. The number of accidents occurring in a plant in a month follows Poisson distribution with mean as 5.2. The probability of occurrence of less than 2 accidents in the plant during a randomly selected month is

A. 0.029

B. 0.034

C. 0.039

D. 0.044

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

6. On an average two covid positive cases are received per day in a hospital. The ratio of the probability of two or less covid positive cases in 2 consecutive days to the probability of exactly two cases on a given day is _______

A. 6.5e–4

B. 6.5e–2

C. 13e–2

D. 13e–4

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

7. If a random variable X has a Poisson distribution with mean 5, then the expectation E[(X + 2)2] equals _______ .

[GATE-2017, 2 MARKS]

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

8. If X is a Poisson’s random variable such that,

P(X = 2) = P(X = 4) + 15 P(X = 6)

Then, standard deviation of X is _______

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

9. If the second moment about origin is 6. The mean of a Poisson's Distribution is ________

A. –3 & 2

B. 2 only

C. –3 only

D. 3 & 2

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

10. Out of 1000 balls 50 are red and rest are white. If 80 balls are picked up at random, the probability of picking up not more than 3 red balls is ______

A.

B.

A normal RV has the PDF of the form

= 1

22 ,

2 is the variance, Var(X)

PDF: x = 1

2σ2 , for all x

P(X ≤ x) = ∞– x 1

2σ2 dx ⇒ can’t be directly integrated

A normal RV ‘’Z’’ is a standard normal RV if μ = 0, σ2 = 1

P.D.F. () =

–

2π e–z

The value of (z) are evaluated by numerical methods and

recorded in tables/calculators.

Var(Y) = a2Var (X)

Key Points

• Area (x ≥ μ) = Area (x ≤ μ) = 1/2

• P x ≥ μ = P x ≤ μ = 1/2

• ∞− μ

1

2

• If X is Normal R.V. having mean = (μ) and variance (σ2). Then

Y = a X + b is also a normal R.V. having the mean = (aμ + b) and

variance = (a2σ2)

Poisson's Distribution and Normal Distribution

• If X is the Normal R.V. having mean = (μ) and variance = (σ2) then

= –

is a standard normal R.V. having mean = 0 and variance = 1

• If X1 and X2 are two independent normal R.V. Then

= 11 + 22 is also a normal R.V. having

mean = a1μ1 + a2μ2 and variance = a1 2 σ1

2 + a2 2 σ2

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Key Points Approximately 68% of the values of any normal distribution

are located within one standard deviation about the mean.

P(μ – σ ≤ X ≤ μ + σ) = 0.6826 ≈ 68.26%

Approximately 95% of the values of any normal distribution

are located within two standard deviation about mean.

P(μ – 2σ ≤ X < μ + 2σ) = 0.9544 ≈ 95.44%

Approximately 99.7% of the value of any normal distribution

are located with in three standard deviation about the mean.

P(μ – 3σ ≤ X ≤ μ + 3σ) = 0.997 ≈ 99.7%

Poisson's Distribution and Normal Distribution

11. A random variable X has normal distribution with mean 100. If P(100 < X < 120) = a then P(X < 80) =

A. 1 – 2a

B. 1 – a

C. 1−2α

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

12. A normal random variable X has the following probability density function

A. 0

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

13. If X is a normal variate with mean 30 and standard deviation 5,

What is the probability P(26 ≤ X ≤ 34) ?

Given A(z > 0.8) = 0.2118.

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

14. A continuous random variable X is normally distributed with mean value 6 and standard deviation 2. Consider the following probabilities, R = P(4 < x < 10) & S = P(0 < X < 8)

Then which of the following is correct

A. R > S

B. R < S

[MSQ]

15. If the masses of 300 students are normally distributed with mean 68 kgs and standard deviation 3 kgs.

(Area under normal curve between z = 0 & z = 1.33 is 0.4082)

A. The number of students whose weight is greater than 72 kg is expected to be 28

B. The number of students whose weight is less than or equal to 64 kg is expected to be 28

C. The number of students whose weight lies between 65 & 71 kg is expected to be 205

D. The number of students whose weight lies between 62 & 74 kg is expected to be 286

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

16. Let X1, X2 be two independent normal random variables with means μ1, μ2 and standard deviations σ1, σ2, respectively. Consider Y = X1 – X2; μ1 = μ2 = 1, σ1 = 1, σ2 = 2. Then,

A. Y is normally distributed with mean 0 and variance 1

B. Y is normally distributed with mean 0 and variance 5

C. Y has mean 0 and variance 5, but is NOT normally distributed

D. Y has mean 0 and variance 1, but is NOT normally distributed

[GATE-2018, 1 MARK]

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

17. Let X be a random variable following normal distribution with mean + 1 and variance 4. Let Y be another normal variable with mean -1 and variance unknown.

If ( ≤ −1) = ( ≥ 2), the standard deviation of Y is

A. 3

B. 2

C. √2

D. 1

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

18. If f(x) = k exp. {–(9x2 – 12x + 13)}, is a p.d.f. of a normal distribution (k, being a constant), the mean and standard deviation of the distribution:

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

19. The lengths of a large stock of titanium rods follow a normal distribution with a mean (μ) of 440 mm and a standard deviation (σ) of 1mm. What is the percentage of rods whose lengths lie between 438 mm and 441 mm?

A. 81.85%

B. 68.4%

C. 99.75%

D. 86.64%

Binomial Distribution Approximation

If be a Binomial R.V. with parameters n & p,

Then for Large values of n, can be approximated as

Normal R.V.

Poisson's Distribution and Normal Distribution

20. Assume that 4 percent of the population over 50 years old has glasses. If a random sample of 3500 people over 50 years is taken, then the probability that less than 150 of them have glasses is _______. (Area under the normal curve between in Z = 0 and Z= 0.86 is 0.3051).

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution

Poisson's Distribution and Normal Distribution