İSTATİSTİK I - Deneme Sınavı - 1

The probability density function f (x) = 0.1 is a straight line above the values 0 to 20, therefore, it is a rectangle with one side being 0.1 value and the other 20. The area is calculated by 20*(0.1) which is equal to 2. The answer is A.

I. 0 ≤ F (x) ≤ 1 (True)

II.  If x₁ ≤ x₂ then F (x₁) ≤ F (x₂) (True)

III.  If x₁ = x₂ then F (x₁) = F (x₂) (False,  If x₁ ≤ x₂ then F (x₁) ≤ F (x₂))

The answer is C.

The mean of an f(x) function is equal to the integral of that function's product to it's respective x on the given range. For the function f(x)= x for 0 ≤ x ≤ 3 the integral of x*f(x) on the range 0 to 4 is taken. This would yield to the integral of x² which is equal to x³/3. For x=0 this would equal to 0, for x=4 this would equal to 9. The mean is equal to 9 - 0 = 9. The answer is B.

The variance is equal to V (X) = E (X²) - [E (X)]² for f(x) = x for 0 ≤ x ≤ 4 the f(x²)=x² which would indicate that x*x² is the function to take the integral in order to find the variance. The integral of x³ is x⁴/4 on 0 ≤ x ≤ 4 would yield to the difference of the value found from x = 4 and x = 0. For x=4 the value is 64 and for x=0 the value is 0. The difference between these values is 64. The answer is E.

For X which is a continuous random variable that is uniformly distributed and takes values between 3 and 9, the minimum value of the random variable X is 3 (it’s the value of a) and the maximum value is 9 (it’s the value of b). Then X~U (a=3, b=9) and the probability density function of the continuous random variable X is defined as, f (x) = 1/(b − a) = 1/(9 − 3)= 1/6 for 3≤x≤9. The answer is B.

For X which is a continuous random variable that is uniformly distributed and takes values between 3 and 9, the minimum value of the random variable X is 3 (it’s the value of a) and the maximum value is 9 (it’s the value of b). Then X~U (a=3, b=9) and the probability density function of the continuous random variable X is defined as, f (x) = 1/(b − a) = 1/(9 − 3)= 1/6 for 3≤x≤9. The mean is equal to (9 + 3)/2 = 6. The answer is E.

For X which is a continuous random variable that is uniformly distributed and takes values between 3 and 9, the minimum value of the random variable X is 3 (it’s the value of a) and the maximum value is 9 (it’s the value of b). Then X~U (a=3, b=9) and the probability density function of the continuous random variable X is defined as, f (x) = 1/(b − a) = 1/(9 − 3)= 1/6 for 3≤x≤9. The variance is equal to (9 - 3)²/2 = 18. The answer is D.

I. The exponential random variable is frequently used to model the time interval between two events. (True)

II. The exponential random variable is defined with a parameter λ. (True)

III. The exponential random variable X defines the time interval between two independent events. (False, the exponential random variable X defines the time interval between two consecutive events.)

The answer is C.

The standard normal curve is symmetric around the mean μ = 0. The standard normal distribution has a standard deviation σ = 1 of the distribution. The area under the standard normal distribution function for P (z ≤ 4.25) is approximately 1.

The answer is C.

A major difference between continuous and discrete random variables is the former takes on uncountable and infinite number of possible outcomes in a given interval. Hence the range of continuous random variable X comprises all real numbers in an interval. In addition to the above given illustrations, water consumption amount in a household, weights of people in a population, the speed of wind in a open certain area, waiting time in a supermarket, checkout lanes or load on a bridge are the few examples for continuous random variable for real world applications. From these examples it’s clear that random variable X can take unaccountably infinite values. To describe such physical structures through continuous random variables density functions are utilized. Therefore, in contrast to discrete random variables, probabilities in continuous random variables can be determined from the area under probability density function (pdf) which is represented by f (x).

Similar to the discrete random variable the mean is a measure of the midpoint or center of the probability distribution.

The variance is a measure of the dispersion or variability for data set for continuous
random variables.

Continuous uniform distribution is the one of the easiest continuous random variable and the probability density function f (x) of the continuous random variable X takes a constant value over the range of the random variable X is defined.

Normal distribution is one of the most significant and extensively used continuous probability
distribution. The major reason for this circumstance is majority of the continuous random variables
which are observed through real life applications (social, medical, physical, biological) are normally or
approximately normally distributed (bell-shaped) variables.

Normal distribution provides basis for the statistical inference. Normal distribution is a symmetric distribution where the random variable values are uniformly scattered around the mean. Population mean, μ and standard deviation σ parameters determine the shape of the normal distribution function. Uniform distribution is different than normal distribution.

Probability density function f (x) of normal distribution has the following properties. 

(i) f (x) ≥ 0 for all x values.
(ii) ∫ ∞  −∞ f (x)dx =1
(iii) The normal distribution function curve is symmetric around the mean, μ.
(iv) The probability density function f (x) does not the touch and intersect x axis.

First property assures that probability density function of random variable x obtain the non-negative values at all times. From the shape of the probability density function curve it’s obvious that pdf, f (x) decreases as the random variable value goes away from the mean, μ. Likewise, probability density function f (x) increases as the random variable value gets closer to the mean, μ. Second property identifies that the area under the probability density function f (x) always equivalent to 1 in the definition interval of the random variable X. Third property suggests that normal distribution curve has a similar shape on both sides of the mean x=μ. That property also proposes the fact that, P (X < μ ) = P (X > μ ) =0.5. Fourth property indicates that the tails of the probability function goes to infinity and at no time crosses or touches the x axis.

The shape of the normal random variable is determined by the mean μ and the standard deviation σ of the distribution.

Exponential distribution is another most significant and extensively used continuous probability distribution. Exponential random variable is frequently used to model the time interval between two events. Exponential random variable is defined with a parameter λ and it’s represented as X~Exponential (λ). In that sense the exponential random variable X defines the time interval between two consecutive events of a Poisson process with a mean of µ = λ. Here λ parameter defines the number of events is a certain time period. Therefore, it’s essential to use consistent time units in the determination of probabilities, mean and variance with the exponential random variable X. The mean (µ) and variance (σ2) for exponential random variable X~Exponential (λ) with parameter λ can be calculated from the following formulas,
E(x) = µ = 1/λ
V (x) =σ2 = 1/λ2

Hence from above given formulas it’s clear that the mean and the standard deviation of the exponential distribution are equal.

Remember on a six-sided fair die, all the outcomes have the same chance to appear, in this example the question makes a restriction of observing values less than 3. There are only two outcomes to satisfy this restriction; those are the numbers 1, and 2; therefore, the probability we are looking for is 

P(less than 3) = 2/6 = 1/3 = 0.333