A random variable X has the following probability distribution: Then is equal to:
Option: 1
Option: 2
Option: 3
Option: 4
Option 2
View Full Answer(11)For three events A, B and C, P(Exactly one of A or B occurs) =P(Exactly one of B or C occurs)
=P(Exactly one of C or A occurs) and P(All the three events occur
simultaneously) Then the probability that at least one of the events occurs, is :
Option: 1
Option: 2
Option: 3
Option: 4
P (exactly one of A or B) =P(exactly B or C)
=P (exactly one of A or C) =
P(A) + P (B)  2P (AB)=
P(B) +P (C)  2P (BC)=
P(C)+P(A)2P (CA)=
P (A) + P (B) 2 P (AB)  P(BC)P(CA) =
P(ABC)=
P(ABC)=
View Full Answer(1)Let A and B be two independent events such that and . Then, which of the following is TRUE?
Option: 1
Option: 2
Option: 3
Option: 4
A & B are independent events
So, A & B' are also independent events, and hence
Correct Option (2)
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Some Important Point Regarding Statistics 
Some Important Point Regarding Statistics
where,
are the means and are the standard deviations of two series.

Old Mean
Old Standard Deviation
If each observation is multiplied with p & then q is subtracted
New mean is the half of old mean
and new standard deviations is the half of old mean
Case 1
Using equation 1
we get
Case 1
Using equation 1
we get
Correct Option (1)
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If the variance of the first n natural numbers is 10 and the variance of the first m even natural numbers is 16, then m+n is equal to _____.
Option: 1 12
Option: 2 18
Option: 3 16
Option: 4 20
Dispersion (Variance and Standard Deviation) 
Variance and Standard Deviation
The mean of the squares of the deviations from the mean is called the variance and is denoted by σ^{2} (read as sigma square).
Variance is a quantity which leads to a proper measure of dispersion.
The variance of n observations x_{1} , x_{2} ,..., x_{n} is given by
Standard Deviation
The standard deviation is a number that measures how far data values are from their mean.
The positive squareroot of the variance is called standard deviation. The standard deviation, usually denoted by σ and it is given by

variance of the first n natural numbers is 10
n = 1,2,3,4........n
Using variance formula
variance of the first m even natural numbers is 16
N = 2,4,6,8........2m
N = 2(1,2,3,4........m)
Using variance formula
Hence
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An unbiased coin is tossed 5 times.Suppose that a variable X is assigned the value k when consecutive heads are obtained for k = 3,4,5, otherwise X takes the value 1. Then the expected value of X, is :
Option: 1
Option: 2
Option: 3
Option: 4
An unbiased coin is tossed 5 times, so total number of outcome is 2^{5} = 32.
Now, the probability for getting number of heads occurring simultaneously
Now expected value is
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Dispersion (Variance and Standard Deviation) 
Variance and Standard Deviation
The mean of the squares of the deviations from the mean is called the variance and is denoted by σ^{2} (read as sigma square).
Variance is a quantity which leads to a proper measure of dispersion.
The variance of n observations x_{1} , x_{2} ,..., x_{n} is given by
Standard Deviation
The standard deviation is a number that measures how far data values are from their mean.
The positive squareroot of the variance is called standard deviation. The standard deviation, usually denoted by σ and it is given by
Variance and Standard Deviation of a Discrete Frequency Distribution
The given discrete frequency distribution be
Variance and Standard deviation of a continuous frequency distribution
The formula for variance and standard deviation are the same as in the case of discrete frequency distribution. Here, is the mid point of each class.
Another formula for Standard Deviation
Shortcut method to find variance and standard deviation
The values of in a discrete distribution or the mid points of different classes in a continuous distribution are large and so the calculation of mean and variance becomes tedious and time consuming.
Here is the shortcut method to find variance and standard deviatio
Let the assumed mean be ‘A’ and the scale be reduced to 1/h times (h being the width of classintervals).
Let the stepdeviations or the new values be .
Replacing x_i from (1) in (2),
Now Variance of the variable x
From (3) and (4),

Correct Option (2)
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Correct Option (1)
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Mean 
Mean of the Ungrouped Data
If n observations in data are x_{1}, x_{2}, x_{3}, ……, x_{n}, then arithmetic mean is given by

Dispersion (Variance and Standard Deviation) 
Variance and Standard Deviation
The mean of the squares of the deviations from the mean is called the variance and is denoted by σ^{2} (read as sigma square).
Variance is a quantity which leads to a proper measure of dispersion.
The variance of n observations x_{1} , x_{2} ,..., x_{n} is given by

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Required probability = when no. the machine has fault + when only one machine has fault + when only two machines have the fault.
Correct Option (4)
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