CH 7 - Statistics PDF

# CHAPTER 7 ## Probability distributions A probability distribution of a random variable can be defined as the relationship between the values that the random variable takes and the probabilities correspond to these values. ### Probability mass function Can be expressed as "for discrete random v...

# CHAPTER 7 ## Probability distributions A probability distribution of a random variable can be defined as the relationship between the values that the random variable takes and the probabilities correspond to these values. ### Probability mass function Can be expressed as "for discrete random variable" a table, formula or graphically. | X | X1 | X2 | ... | Xn | | --------- | -------- | -------- | ----- | -------- | | P(x) | P(x1) | P(x2) | ... | P(xn) | The prob. mass function must satisfy two conditions: 1. 0<P(x)<1 2. Σp(xi) = 1 **Example:** Find the values of random variable *x* where *x* represent the number of heads when three coins are thrown and find the prob. mass function. And find: 1. P(x≥1) 2. P(1≤x<3) 3. P(x<2) **Ans.** S= {HHH, HHT, HTH, THH, TTT, TTH, THT, HTT} | X | Available values of x | | - | --------------------- | | 0 | | | 1 | | | 2 | | | 3 | | **CHAPTER 7** | **P.M.F** | X | P(x) | | ------------- | - | ------ | | | 0 | 1/8 | | | 1 | 3/8 | | | 2 | 3/8 | | | 3 | 1/8 | 1. 0<P(x)≤1 2. ΣP(x)= 1 **Mathematical Formula:** * P(x) = ```1/8``` : x = 0, 3 * P(x) = ```3/8``` : x = 1, 2 * P(x) = ```0``` : Otherwise **Graphically** | | | -------------- | | **P(x≥1) = P(x=1) + P(X=2) + P(x=3)** | | **(3/8 + 3/8 + 1/8 = 7/8)** | | **P(1≤x < 3) = P(x=1) + P(x = 2)** | | **(3/8 + 3/8 = 6/8)** | | **P(x<2) = P(x = 0) + P(X = 1)** | | **( 1/8 + 3/8 = 1/8 = 0.5)** | **Example:** The following table represents the Probability mass function for random variable *X* | X | P(x) | | - | ----- | | 4 | 0.12 | | 5 | 0.10 | | 7 | 0.4 | | 9 | k | | 12 | 0.2 | Find the value of k and P(5≤x≤9) **Ans.** ΣP(x) = 1 Then, 0.12 + 0.10 + 0.4 + k + 0.2 = 1 0.82 + K = 1 K = 0.18 P(5≤x≤9) = 0.1 + 0.4 + 0.18 = 0.68 The **mean** (expected value) for probability mass function: μ = E(x) = ΣxP(x) | X | X1 | X2 | ... | Xn | | --------- | -------- | -------- | ----- | -------- | | P(x) | P(x1) | P(x2) | ... | P(xn) | | xP(x) | x1P(x1) | x2P(x2) | ... | xnP(xn) | ΣxP(x) = x1P(x1) + x2P(x2) + ... + xnP(xn) ### The Variance and Standard deviation V(x) = E(x²) - μ² Where E(x²) = Σx²p(x) S.D(x) = √V(x) **Example:** From the following prob. mass function find the mean and variance: | X | P(x) | | --- | ----- | | 2 | 0.1 | | 5 | 0.13 | | 7 | 0.3 | | 8 | 0.27 | | 10 | 0.2 | **Ans.** μ = E(x) = Σxp(x) = (2 x 0.1) + (5 x 0.13) + (7 x 0.3) + (8 x 0.27) + (10 x 0.2) = 7.11 μ = 7.11 Variance V(x) = E(x²) - μ² E(x²) = Σx²p(x) = (2² x 0.1) + (5² x 0.13) + (7² x 0.3 ) + (8² x 0.27) + (10² x 0.2) = 55.63 E(x²) = 55.63 V(x) = (55.63) - (7.11)² = 5.07 Standard deviation SD(x) = 2.253 **Example:** From the following P.M.F find value of k and c. | X | P(x) | | - | ----- | | 2 | 0.2 | | 4 | 0.4 | | c | 0.3 | | 7 | k | If the value of mean = 4.2 **Ans.** The prob. mass function then ΣP(x) = 1 0.2 + 0.4 + 0.3 +k = 1 0.9 + k = 1 k = 0.1 E(x) = ΣxP(x) = (2 x 0.2) + (4 x 0.4) + (0.3c) + ( 7 x 0.1) = 4.2 2 + 1.6 + 0.3c + 0.7 = 4.2 0.3c = 1.5 c = 5 ## Probability density Function Can be expressed as "For Continuous Random Variable" formula or graphically but not as table. The probability density function must satisfy the two conditions: 1. f(x) > 0 2. ∫f(x)dx = 1 **Example:** Prove that the following function is P.d.f. *f(x) = 1/9 *x*² 0≤x≤3 *f(x) = 0* Otherwise And find P(x≥2), P(1≤x<3), P(X=1) **Ans.** * The mean and Variance for prob. density functions The mean (expected value) μ = E(x) μ = ∫f(x)dx The Variance V(x) = E(x²) - μ² Where E(x²) = ∫x² f(x) dx S.D(x) = √V(x) **CHAPTER 7** **Example:** Find the value of k and find the mean and Variance. f(x) = kx² 0 ≤ x ≤ 3 f(x) = 0 Otherwise **Ans.** P.d.f Then ∫f(x)dx = 1 ∫kx² dx = 1 K = 1/3 Then f(x) = (1/9)*x² 0 < x <3 f(x) = 0 Otherwise The mean μ = ∫xf(x)dx= ∫x ( 1/9 *x*² ) dx = 2.25 V(x) = E(x²) - μ² E(x²) = ∫x² (f(x)) dx = ∫x² (1/9 *x*² ) dx = 5.4 V(x) = 5.4 - (2.25)² = 0.3375 **Example:** Given the following PDF for RV *y* f(y) = ky 0 ≤ y ≤ 4 0 Otherwise Find the value of k and find P(2.5≤Y ≤3.5), P(y=2), P(y>2) and find the mean and Variance **Some Special Probability distributions:** ### **Binomial Distribution:** The binomial dist? is one of the most widely encountered prob. distn in the applied statistics and is derived from a process known as the Bernoulli trail. **The Conditions of binomial experiment:** 1. The experiment is repeated *n* times (trails). 2. Each trial has only two outcomes (success or failure). 3. The prob. of success remains constant in each trail and denoted by *p* and prob. of failure by *q = 1 - p*. 4. The *n* trails are independent **CHAPTER 7** The prob. mass function for binomial distn P(x = x) = Cx pxqn-x x: 0 1 2 … n The mean for binomial dist? E(x) = M = np Variance of the binomial distn V(x) = σ² = npq Note that: (V.V.V. Important) * at least a P(x≥a) * more than a (greater than a) P(x > a) * at most a P(x ≤ a) * less than a (fewer than a) P(x < a) * b/w a and b P(a < x < b) * b/w a and b, inclusive P(a ≤ x ≤ b) **Complement rule:** (V.V.V. Important) * P(x ≥ a) = 1 - P(x < a) * P(x > a) = 1 - P(x ≤ a) * P(x ≤ a) = 1 - P(x > a) * P(x < a) = 1 - P(x ≥ a) **Example:** Suppose that in a certain population of men, the probability that a man has high blood pressure is 0.15. If a sample of 6 men is selected randomly from this population find: 1. The prob. distribution of the number of men with high blood pressure 2. The mean and the variance of the number of men having high blood pressure 3. The Prob. that the selected sample contains three men having high blood pressure 4. The prob. that the selected sample contains at least one man with high blood pressure. **Example (2):** Melchart et al. (2005) give the successful response rate to acupuncture treatment in 124 patients with tension type headache as 46%. (From their data we have P = 58/124 =0.46). Suppose a doctor treated four acupuncture patients. What is the prob. that at most one responds? **Example:** Given a family having three children with equal probabilities of child gender [male and female] write the p.m. f of the number of female children (girls) in this family, then find: * The mean and the variance of the number of girls in this family * The prob. that the family includes exactly two girls? * The prob. that the family includes at least two girls? * The prob. that the family includes at most two girls. **Example:** Suppose it's known that the probability of recovery from a certain disease is 0.2. If 10 people are stricken with disease. What is the prob. that: *8 or more from them will recover? *fewer than three from them will recover? **Example:** The prob. that a patient recovers from a stomach infection is 0.3. Suppose that ten patients are known to have contracted this infection. What is the probability that: *Exactly seven from them will recover? *At least two from them will recover? *At most eight from them will recover? **CHAPTER 7** **Example:** 20% of the people in a large population suffer from high blood pressure. Two people were randomly drawn from this population. What's the probability that: *Both of them suffer from high blood pressure *At least one of them suffer from high blood pressure *None of them suffers from high blood pressure. *Exactly one of them suffer from high blood pressure? **Example:** In a couple where each person is heterozygous for the sickle-cell gene, there is a probability 0.25 that any child of the couple will actually have disease. In randomly selected family of 6 children where both patients have the disease: *What's the prob. that 2 children have the disease? *What's the prob. that at most one child has the disease? **Example** Suppose it is known that 30% of a certain population are immune to some disease. If a random sample of size 10 is selected from this population. Find that the prob. at least two have immune from the disease. **Example:** Mid-term 2018 Let x be a random variable which has a binomial dist with M =12 and σ =3. Find the parameters n, p. ## Poisson Distribution: The Poisson distribution is used to describe discrete quantitative data. Such as counts that occur independently and randomly in time or space at some average rate. For ex. *The number of deaths in a town from a particular disease per day *The number of admissions to a particular hospital *Weekly traffic accidents in a town The P.M.F for Poisson distn P(x=x) = (e^-λ * λ^x)/x! x : 0 1 2…. **CHAPTER 7** * The mean and variance: μ = λ Variance σ² = λ Only in Poisson distn mean = λ = Variance = λ = parameter σ = √λ Standard deviation = √λ Note that: P(x ≥ a) = P(x=a) + … ∞ = 1 - P(x < a) P(x>a) = P(x=a+1) + … ∞ = 1 - P(x ≤ a) **Example:** Suppose that the number of snake bite cases treated in Benghazi Medical Center (BMC) during a certain year has the Poisson dist? with average 6 bite cases. What's the probability that: 7 snake bite cases are treated in BMC during this specific year? Less than 2 snake bite cases are treated in BMC during this specific year? 10 snake bite cases are treated in BMC during two years under the same conditions. No snake bite cases are treated in BMC during a month in this specific year. **Example:** The switchboard ring at a service desk receives, on average, five calls in ten minutes. What's the prob. that during a ten-minute interval the service desk will receive: *No more than three calls? *At least two call? *Exactly five calls? **Example:** Suppose that the number x of cases visiting the emergency clinic at Benghazi Medical Center is two cases per day. Then the probability: *At least one case will be visiting the emergency clinic at any day is equal to… *Exactly 5 cases will be visiting the emergency clinic at the next week is equal to… **Example:** In a certain population 13 new cases of esophageal cancer are diagnosed an average each year. If we assume that the incidence esophageal cancer follow Poisson dist? what's prob. that in a given year the number of newly diagnosed will be 10. **CHAPTER 7** **Example:** A life insurance salesman sells, on average, 3 life insurance policies per week. *What's prob. he will sell some policies per week? *Assuming that there are five working days per week, what's the prob. that in a given day he will sell one policy? **Example:** If y is a random variable following Poisson dist such that: *Y~ p(x) P(y = 1) = 3*P(y = 0)* *(a) the mean and the standard deviation of y are… *(b) the prob. that y lies the interval [4,7] is… **Normal (Gaussian) distribution** Continuous distn in statistics because it is the most many phenomena in statistics are normally or approximately normally distributed e.g high, weight, age The normal distn dependent upon two parameters: mean μ and variance σ² f(x) = 1/√(2π σ²) * e ^-1/2(x-μ/σ²) -∞ < x < +∞ -∞ < μ < +∞ σ² > 0 ### The properties of Normal Distr. 1. The normal curve has a bell-shaped curve. 2. The normal curve symmetric about the mean. (mean = median = mode) 3. The normal dist? depends upon two parameters: μ mean and variance σ² The location of the curve depends upon the mean. The shape of the curve depends upon the variance. **CHAPTER 7** 4. The total area under the curve = 1. * b/w μ-0, μ+0 = 0.6826 68.26% * b/w μ-2σ, μ+2σ = 0.9544 95.44% * b/w μ-3σ, μ+3σ = 0.9974 99.74% ### Standard Normal Distribution : Special Case of *the* Normal distn if μ = 0 and σ =1. The rv following the standard Normal dist. is Z ~ N(0,1). The standard normal distn is very important because it helps in calculating the probability of any normal distn by using: Z = x - μ / σ Calculating the probabilities using p.d.f. of normal dists. is very difficult but can be easily by using a table of standard normal dist (z-table). Note that: If x Normal random variable with mean μ and variance σ² can be converted to standard normal random variable z with mean 0 and variance 1 by using: Z = x - μ / σ P(z ≤ a) = P(z < a) = From Table P(z ≥ a) = P(z > a) = 1 - P(z ≤ a) P (a < z < b) = P(z < b) - P(z < a) **Example:** The weights of a population of pregnant women is distributed normally with mean 55 and variance 9 kg. A pregnant woman has been randomly chosen from this population. What is the probability: *Less than 60 kg *More than 53 kg *B/w 50 kg and 75 kg **Example:** Brain weights of male elephants are approximately normally distributed with mean 1.4 kg and variance 0.11 kg. Find the approximate proportion of male elephants with a brain weight: *B/w 1.5 and 1.6 kg *B/w 1.2 and 1.3. Explain the link b/w answers **Example:** Suppose that in a certain pediatric population, casual sitting systolic blood pressure is normally distributed with mean 115 and variance 225. *The prob. that a child randomly selected from this population will have systolic pressure at least 140… *The prob. that a child randomly selected from this population will have systolic pressure b/w 100 and 130. **Example:** The grades of a group of 2000 students in an exam are normally distributed with a mean of 60 and a standard deviation of 5. A student from this group is selected randomly. *The probability that his/her grade (a randomly selected in an exam) is greater than 70 *The probability that his/her grade is less than 55 *Approximately how many students have grades b/w 55 and 70. **CHAPTER 7** **Example:** (2015-2016, (MCQ) *After studying a couple's family history, a doctor determines that the probability of any child born to this couple having a gene for disease is 0.25. If this couple has three children. Then:* *The prob. that at least one of the children have the gene for disease is equal to… *The mean and variance of the number of children have the gene for disease is equal to…. **Example:** Suppose that the number of snake bite cases treated in Benghazi Medical Center (BMC) during a certain year has the Poisson dist? with average 6 bite cases, what’s the probability that: *7 snake bite cases are treated in BMC during this specific year? *Less than 2 snake bite cases are treated in BMC during this specific year? *10 snake bite cases are treated in BMC during two years under the same conditions. *No snake bite cases are treated in BMC during a month in this specific year. **Example:** The switchboard ring at a service desk receives, on average, five calls in ten minutes. What’s the prob. that during a ten-minute interval the service desk will receive: *No more than three calls? *At least two call? *Exactly five calls? **Example:** Suppose that the number x of cases visiting the emergency clinic at Benghazi Medical Center is two cases per day. Then the probability: *At least one case will be visiting the emergency clinic at any day is equal to… *Exactly 5 cases will be visiting the emergency clinic at the next week is equal to… **Example** In a certain population 13 new cases of esophageal cancer are diagnosed an average each year. If we assume that the incidence esophageal cancer follow Poisson dist? what’s prob. that in a given year the number of newly diagnosed will be 10.